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IN  MEMORIAM 
FLORIAN  CAJORl 


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Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/completearithmetOOmaglrich 


/ V| r^n  U-y,,  H--  -f   7t'-^t^*f 


Greenleaf* s  Mathematical  Series. 


THE 


COMPLETE 

AEITHMETIC, 

ORAL   AND    WRITTEN, 


ON   tritE, BASIS    OF    WOJlJiS" 

By  BETJJAMm  ^GEEEKLE 


LEACH,    SHEWELL,   AND   SAKBOEK, 

bosto:n^  and  new  york. 


GREENLEAF'S 
MATHEMATICAL    SERIES. 


INDUCTIVE    COURSE. 

First  Lessons  in  Numbers. 

A  Brief  Course  in  Arithmetic. 

The  Complete  Arithmetic. 

The  Brief  Course  and  the  Complete  Arithmetic  are 
each  published  wiib,^^nd  withoiit-f^Kswers. 

Key  to  T,HEJGoMl^Ll!r^¥'/A^•lTtIMteTIC,  for  Teachers  ordy^ 


CAJORf 


Copyright,  18S1,  By  Henry  B.  Maglathlik. 


pBSttwoxK  BY  Berwick  &  Smith,  Boston,  U.S.A. 


PEEFACE. 


oi^o 


This  Arithmetic,  undertaken  at  the  suggestion  of  many 
educators  of  distinction,  has  been  prepared  with  special  refer- 
ence to  training  for  practical  business,  and  to  development  of 
mind  power  through  fixed  habits  of  attention  and  lucid  pro- 
cesses of  reasoning. 

To  secure  skill,  rapidity,  and  accuracy  in  the  use  of  numbers, 
required  in  common  transactions,  a  large  number  of  compara- 
tively simple  business  questions  has  been  provided,  and  promi- 
nence has  been  given  to  subjects  of  the  most  practical  value. 

That  useful  mental  discipline  may  be  attained  the  theory  and 
principles  of  numbers  have,  been  clearly  presented,  and  problems 
have  been  given  requiring  thought  and  discrimination. 

The  inductive  plan  has  been  followed  throughout,  principles 
have  been  developed  from  methods,  rules  derived  from  analyses, 
and  oral  and  written  exercises  combined  in  a  rational  manner. 

The  greatest  care  has  been  observed  to  have  the  definitions 
brief,  clear,  and  accurate,  and  the  solutions  simple,  concise,  and 
logical. 

The  methods  employed  are  those  which  business  experience, 
or  test  in  the  school-room,  has  shown  to  be  the  best. 

Decimals  to  three  places,  and  United  States  money,  are  simply 
treated  at  the  beginning  with  integers. 

The  problems  are  abundant  and  varied,  based  on  recent  and 
reliable  data,  and  drawn  from  the  acjiml  experiences  of  life.    In 


wn  from  the  actual 

9183iSr 


IV  PREFACE. 

commercial  arithmetic  the  usages  of  the  best  business  houses 
have  been  followed. 

Several  hundred  examples  which  have  been  used  in  exami- 
nations by  superintendents  of  public  instruction  in  various  cities 
and  towns  leading  in  educational  matters,  have  been  collected, 
and  arranged  as  exercises  for  testing  proficiency,  and  for  supple- 
mentary  practice,  to  be  drawn  from  at  the  teacher's  discretion. 

Much  matter  formerly  considered  necessary  in  an  arithmetic, 
but  which  modern  progress  has  rendered  useless  or  antiquated, 
has  been  omitted. 

The  Appendix  contains  tables  for  reference ;  information  of  a 
somewhat  technical  nature  for  the  business  man,  the  mechanic, 
and  the  farmer;  subjects  of  minor  importance  to  the  majority 
of  pupils ;  and  rules  and  applications  not  needed  in  the  body  of 
the  work. 

Indebtedness  is  gratefully  acknowledged  to  school  superinten- 
dents of  vanous  cities  and  towns  for  examination  papers  and 
valuable  suggestions ;  to  the  Metric  Bureau  for  information  and 
cuts ;  to  Boston  Custom-House  officials  for  copies  of  invoices  ; 
to  the  Kegents  of  the  University  of  the  State  of  New  York, 
and  to  various  college  authorities,  for  entrance  test-papers  to  be 
found  in  the  Appendix. 

Credit  is  due  to  all  the  able  teachers  who  have  aided,  dur- 
ing the  past  three  years,  in  the  preparation  of  the  Inductive 
Course,  of  which  this  book  is  a  part.  The  larger  share  of  this 
credit  belongs  to  Mr.  G.  A.  South  worth,  the  successful  and 
experienced  Master  of  the  Prescott  Grammar  School,  Somer- 
ville,  Mass.,  by  whom  much  work  has  been  done.  His  prac- 
tical knowledge  of  what  both  teachers  and  pupils  need  in  a 
text-book  has  been  of  great  advantage. 

The  work  is  given  to  the  public  in  the  expectation  that  it 
will  meet  every  reasonable  requirement  of  our  common  schools 
and  seminaries. 


CONTENTS. 


Notation  and  Numeration 1 

Addition 11 

Subtraction 20 

Miscellaneous  Exercises 26 

Multiplication  27 

Miscellaueous  Exercises 36 

Division 38 

Miscellaneous  Exercises 47 

Review 49 

Factors 54 

Cancellation 56 

Greatest  Cornniou  Divisor. .  57 

Least  Common  Multiple 60 

Miscellaneous  Exercises 62 

Common  Fractious 63 

Keduction  of  Fractions 65 

Addition  of  Fractions 73 

Subtraction  of  Fractions  ...  74 

Multiplication  of  Fractions.  76 

Division  of  Fractions 81 

Miscellaneous  Exercises 89 

Review 91 

Decimal  Fractions 95 

Reduction  of  Decimals 97 

Multiplication  of  Decimals.  100 

Division  of  Decimals 101 

Miscellaneous  Exercises 104 

United  States  Money 106 

Aliquot  Parts 109 

Accounts  and  Bills 112 

Weights  and  Measures 116 

Length  Measures 116 

Surface  Measures 117 

Volume  Measures 118 


Capacity  Measures 119 

Weights 120 

Time 121 

Arc  and  Angle 123 

Miscellaneous 124 

Compound  Numbers 126 

Reduction 126 

Addition 134 

Subtraction 136 

Multiplication 138 

Division 138 

Miscellaueous  Exercises —  139 

The  Metric  System 141 

Length  Measures 142 

Surface  Measures 143 

Volume  Measures 1 44 

Capacity  Measures 145 

Weight  Measures 146 

Reduction  of  Units 148 

Measurements 151 

Surfaces 161 

Volumes 154 

Wood  Measure 157 

Board  Measure 158 

Miscellaneous  Exercises... .  159 

Review * 161 

Percentage 167 

Profit  and  Loss 173 

Commission 175 

Insurance 177 

Miscellaneous  Exercises —  178 

Interest 181 

Simple  Interest 181 

Exact  Interest 189 


VI 


CONTENTS. 


Problems  in  Interest 190 

Partial  Payments 194 

Compound  Interest 199 

Discount 203 

True  Discount 203 

Commercial  Discount 205 

Bank  Discount 206 

Miscellaneous  Exercises 210 

Stock  Investments 212 

Exchange 216 

Domestic  Exchange 217 

Foreign  Exchange 219 

Average  of  Payments 222 

Keview 225 

Ratio  and  Proportion 228 

Ratio 228 

Proportion 230 


Simple  Proportion 232 

Compound  Proportion 234 

Partnership 237 

Involution  and  Evolution 241 

Involution 241 

Evolution 242 

Square  Root 243 

Cube  Root 249 

Mensuration  256 

Right-angled  Triangles  256 

Quadrilaterals  257 

Prisms  260 

Pyramids  and  Cones 260 

Similar  Surfaces  263 

Similar  Solids 264 

Review 265 

Examination  Questions 273 


Appendix. 


Roman  Notation 308 

Fundamental  Processes 309 

Prime  Numbers 310 

Circulating  Decimals 310 

Tables 312 

Government  Lands 313 

Longitude  and  Time   314 

Legal  Interest 316 

Twelve  per  cent  Interest    317 

Animal  Interest 318 

Aveiage  of  Accounts 321 

Business  Forms 323 

Taxes 325 


Duties 327 

Measurement  of  Round  Timber  329 

Gauging    330 

Tonnage  of  Vessels... 331 

Farmers'  Estimates 332 

Stone  and  Brick  Work 333 

Builders'  Estimates 335 

Arithmetical  Progression    337 

Geometrical  Progression 339 

College  Entrance-Examination 

Papers 341 

Answers  to  Examples 347 


THE 


COMPLETE    ARITHMETIC. 


1.  A  Unit  is  a  single  thing,  or  one ;  as  one  hour,  one 
dollar,  one. 

2.  A  Number  is  a  unit,  or  a  collection  of  units ;  as  one 
montli,  four  hours,  seven. 

3.  Arithmetic  treats  of  numbers  and  their  use. 

NOTATION    AND    NUMERATION. 

4.  Notation  is  writing  numbers  in  figures. 

5.  Numeration  is  reading  numbers  written  in  figures. 

6.  Figures  are  characters  used  to  express  numbers. 
Ten  different  figures  are  used ;  they  are 

0,       1,       2,       3,       4,       5,       6,       7,       8,       9. 

Zero,     One,     Twc,    Three,    Four,    Five,      Six,    Seven,   Eight,   Nine. 

7.  Zero,  or  cip/ieVy  used  alone   expresses  no  units,  oi 
nothing. 

The  other  nine  figures  express   the   number   of  units 
shown  by  their  names. 

8.  To  express  numbers  larger  than  nine,  two  or  more 
figures  are  written  side  by  side. 


2  NOTATION  AND  NUMERATION. 

*  9.  A  figure  written  alone  has  only  a  simple  name  and 
value;  but  when  used  with  other  figures  it  has  also  a 
place-name  and  value. 

10.  The  Place  of  a  figure  is  its  position  with  reference 
to  another  figure  in  a  number. 

11.  When  two  figures  are  written  side  by  side,  the  fig- 
ure at  the  right  occupies  the  first  jjlace,  has  the  place-name 
ones,  and  expresses  iinits  of  the  first  order.  The  figure  at 
the  left  occupies  the  secoTul  place,  has  the  place-name  tens, 
and  expresses  units  of  the  second  order,  each  of  which  is 
ten  tinges  larger  than  a  unit  of  the  first  order.     Thus, 

80  is  read  8  tens,  0  ones,  or  briefly  eighty. 

75       "       7  tens,  5  ones,         "         seventy-five. 

By  the  use  of  the  other  possible  combinations  of  two 
figures  any  number  between  nine  and  one  hundred  may  be 
written. 


Eead  the  following : 

1.   97            4. 

84 

7.   54 

10.   48 

2.    45               5. 

77 

8.   91 

11.   89 

3.   63            6. 

38 

9.   26 

12.   98 

Write  in  figures : 

13.  Eighty-six.       16.  Thirty-six.  19.  Eighty-one. 

14.  Sixty-four.        17.  Four  tens,  eight  ones.  20.  Twenty-nine. 

15.  Seventy-nine.  18.  Ninety-two.  21.  Sixty-seven. 

12.  A  figure  at  the  left  of  the  tens*  figure  occupies 
the  third  place,  has  the  place-name  hundreds,  and  ex- 
presses a  third  order  of  ttnits,  each  ten  times  as  large 
as  cue  of  those  in  the  second  place.     Thus, 

659  is  read  6  hundreds,  5  ten^,  9  ones,  or  briefly,  six 
hundred  fifty-nine.  508  is  read  5  hundreds,  0  tens,  8  on^es, 
or  briefly,  five  hundred  eight. 


NOTATION   AND    NUMERATION. 


By  using  the  different  combinations  of  three  figures  any 
number  between  ninety-nine  and  one  thousand  may  be 
expressed. 


Eead  the  following 

: 

22.  789      25. 

123 

28. 

656 

31. 

555 

23.  456      26. 

984 

29. 

700 

32. 

390 

24.  804      27. 

327 

30. 

608 

33. 

299 

Write  in  figures : 

34.  Eight  hundred  forty-four.  38.  Five  hundred  eighty-eight. 

35.  Two  hundreds,  seven  ones.  39.  Seven  hundred. 

36.  Seven  hundred  fifty-three.  40.  Six  hundred  sixty-six. 

37.  Nine  hundred  ninety.  41.  Nine  hundred  four. 

13.  A  figure  at  the  left  of  the  hundreds'  figure  occu- 
pies the  fourth  place,  has  the  place-name  thousands,  and 
expresses  the  fourth  order  of  units,  each  ten  times  as  large 
as  one  in  the  third  place.     Thus, 

8976  is  read  8  thousands,  9  huridreds,  7  te7is,  6  ones,  or 
briefly,  eight  thousand  nine  hundred  seventy-six. 

In  this  way  the  various  combinations  of  four  figures 
may  be  used  to  express  the  numbers  between  nine  hundred 
ninety-nine  and  te7i  thousand, 

Eead  the  following : 

42.  7382  45.  9308  48.  1333 

43.  8641  46.  4629  49.  3004 

44.  6047  47.  5432  50.  2908 

Write  in  figures : 

51.  Six  thousands,  no  hundreds,  five  te7is,  seven  ones. 

52.  Eight  thousands,  two  hundreds,  no  tens,  no  ones. 

53.  How  may  the  preceding  number  be  read  more  briefly  ? 

54.  Three  thousand  eight  hundred  forty-six. 

55.  Nine  thousand  two  hundred  seven. 


4  NOTATION    AND    NUMEKATION. 

56.  Five  thousand  seven  lumdred  forty. 

57.  Two  thousand  eight. 

58.  Four  thousands,  four  tens,  four  ones. 

59.  Eight  ones,  three  tens,  nine  hundreds,  seven  thousands. 

14.  On  the  same  principle  larger  numbers  are  writ- 
^^en  by  using  five,  six,  seven,  eight  or  more  figures,  each 
additional  figure  filling  a  new  place,  named  successively 
towards  the  left,  ten-thousands,  hundred-thousands,  millions, 
ten-millions,  and  so  on,  and  forming  a  new  order  of  units, 
each  of  which  is  ten  times  as  large  as  the  next  smaller. 

15.  The  2)lcice-names  and  values  of  figures  are  shown  in 


the  following 

TABLE. 

Place. 

Place-name. 

Order  of  Units. 

First, 

Ones, 

1st, 

Second, 

Tens, 

2d, 

Third, 

Hundreds, 

3d, 

Fourth, 

Thousands, 

4th, 

Fifth, 

Ten-thousands, 

5th, 

Sixth, 

Hundred-thousands, 

6th, 

Seventh, 

Mimons, 

7th, 

Eighth, 

Ten-millions, 

8th, 

Ninth, 

Hundred-millions, 

9th, 

Tenth. 

Billions. 

lOLh. 

16.  For  convenience  in  reading  and  writing  numbers, 
each  three  places  in  a  number,  beginning  with  the  ones, 
form  a  group.  Each  group  is  named  from  its  right-hand 
order  of  units. 

The  first      group  is  the  group  of  ones. 
"     second  "  "  "         thousands, 

"     third  "  "  "         niillions. 

"    fotcrth  "  "  "         unions. 

The  comma  is  used  to  mark  oil'  the  groups. 


NOTATION   AND   NUMERATION.  5 

Thus,  in  the  number  109,876,543,210, 

the  first      group  is  210  ones, 
the  second  group  is  543  thousandSy 
the  third    group  is  876  millions, 
the  fourth  group  is  109  billions. 

17.  Each  complete  group  contains  hundreds,  tens,  and 
07ies  of  its  group-name.     Thus, 

847,298,341,  instead  of  being  read  8  hundred-millions, 
4  ten-millions,  7  millions,  etc.,  is  more  briefly  read,  eight 
hundred  forty-seven  millions,  two  hundred  ninety-eight 
thousands,  three  hundred  forty-one. 

EXERCISES. 

60.  In  the  number  321,654,987,  what  is  the  name  of  the 
first  group  ?     Of  the  third  ?     Of  the  second  ? 

61.  Give  the  place  name  of  the  1.  Of  the  2.  Of  the  3. 
Of  the  5.     Of  the  9.     Of  the  6.     Of  the  4. 

62.  What  figure  expresses  the  4th  order  of  units  ?  The 
7th?     The  5th?     The  6th?     The  8th? 

18.  The  method  of  writing  numbers  in  figures  is  based 
on  the  following 

Principles  of  Notation. 

1.  Ten  units  of  any  order  equal  one  unit  of  the  order 
next  larger, 

2.  Each  removal  of  a  figure  one  place  toward  the  left  in- 
creases its  value  ten  times,  and  each  removal  of  a  figure 
toward  the  right  diminisftes  its  value  ten  times, 

19.  Our  system  of  Numbers  is  called  a  Decimal  System, 
from  the  Latin  decem,  ten,  because  of  the  uniform  ten-fold 
increase  of  units  from  any  order  to  the  next  larger. 

The  successive  orders  of   units   in   a   number  form  a 


6  NOTATION   AND   NUMERATION. 

Scale ;  and,  where  ten  units  of  any  order  always  make  a 
unit  of  the  next  larger,  the  scale  is  ten  or  decimal. 

20.  A  period  (.),  called  the  decimal  pointy  may  be  writ- 
ten at  the  right  of  the  ones  figure  to  m'ark  the  ones'  place. 

21.  A  number  at  the  left  of  the  decimal  point  is  an 
Integer,  or  whole  number,  because  it  is  a  collection  of  en- 
tire  ones. 

22.  Figures  may  be  written  at  the  right  of  ones,  if 
separated  from  them  by  the  decimal  point. 

23.  The  first  figure  at  the  right  of  ones  is  in  the  first 
decimal  place,  has  the  place-name  tenths,  and  expresses 
the  first  order  of  decimal  units,  each  being  one  tenth  of  a 
one. 

A  figure  at  the  right  of  tenths  is  in  the  second  decimal 
place,  has  the  place-name  hundredths,  and  expresses  the 
second  order  of  decimal  units,  each  being  one  tenth  of  a 
tenth. 

A  figure  at  the  right  of  hundredths  is  in  the  third  deci- 
mal place,  has  the  place-name  thousandths,  and  expresses 
the  third  order  of  decimal  units,  each  being  one  tenth  of 
a  hundredth.     Thus, 

6.378  expresses  six  ones,  and  three  tenths,  seven  hun- 
dredths, eight  thousaTidtlis,  or  briefly  six,  and  three  hun- 
dred seventy-eight  thousandtJis. 

Note.  —  In  reading  numbers  use  the  conjunction  and  only  after  the  ones  to 
show  the  place  of  the  decimal  point 

24.  A  number  at  the  right  of  the  decimal  point  is  a 
Decimal,  or  a  collection  of  tenths,  hundredths,  thousandths 
or  other  decimal  parts  of  a  one, 

I40TB.  —  In  writing  a  decimal  without  an  integer  the  ones'  place  may  hA 
filled  with  a  zero.    Thus, 

Three  hundred  forty-one  thousandths  may  be  wntten  0.341. 


NOTATION   AND   NUMERATION.  7 

25.    The  method  of  writing  numbers,  and  the  names  of 
units,  places,  and  groups  are  shown  in  the  following 


TAJBLE. 


Integer. 


Decimal. 


ORDERS  OF  Units.  12th,Uth,10th,  9th,  8th,  7th.   6th,  5th,  4th,    3d,    2d,    Ist,         Ist,  2d.   3d. 


§ 


Place-names. 


Figures. 


Groups. 
Group-names. 


c3 


nfj    ;^  ""g    ■— {     ^^      "T? 


rO 


^       r^ 


3   6 


,5 


0,79 


i 


4th, 
Billious, 


3d,  2d, 

Millions,      Thousands, 


5  < 

1st, 
Ones, 


^    1 


6  3   8 


1st  Decimal, 
Thousandths. 


The  number  in  the  table  is,  three  hundred  sixty-one 
billions,  five  hundred  forty  millions,  seven  hundred  ninety- 
three  thousands,  one  hundred  fifty-four,  and  six  hundred 
thirty-eight  thousandths. 

Note  1.  —  Group-names  above  billions  are  trillions,  quadrillions,  quintillions, 
etc.  ;  below  thousandths,  millionths,  billionths,  etc. 

Note  2.  —  The  left-hand  group  of  the  integer,  or  the  right-hand  group  of  the 
decimal  may  contain  only  one  or  two  figures.    The  name  of  a  partial  decimal 
group  is  the  same  as  the  place-name  of  its  right-hand  figure.    Thus, 
16.84  is  read  sixteen  and  eighty-four  hundredths. 


EXERCISES    IN    WRITING    NUMBERS. 

63.  Write  in  figures,  thirty-five  million  six  hundred 
twenty-seven  thousand  two  hundred  three,  and  sixteen  hun- 
dredths. 

Solution.  —  Writing  35  for  the  millions,  627 
ot^  ^orr  oAo  1 /»      foi"  tliB  thousands,  203  for  the  ones,  and  .16  for 
the  hundredtbs,  we  have  as  the  result  required, 
35,627,203.16. 


8  NOTATION   AND   NUMERATION. 

64.  Write  in  figures,  three  hundred  twenty-nine  thousand 
four  hundred  fifteen. 

65.  Write  in  figures,  nine  thousand  seven  hundred  fifty- 
two,  and  one  hundred  one  thousandths. 

26.     Rule  for  writing  Numbers. 

Beginning  at  the  lefty  write  the  figures  of  each  group  in 
their  order,  filling  vacant  xjlaces  and  groups  with  ciphers. 

Write  in  figures: 

66.  Six  thousand  five  hundred  nine. 

67.  Eleven  thousand  nine  hundred  eleven. 

68.  Thirty-seven  thousand  four  hundred  eighty-nine. 

69.  Ninety  thousand  four  hundred  forty- four. 

70.  One  hundred  sixty-three  thousand. 

71.  Two  hundred  twenty  thousand  two  hundred  sixty-two, 

72.  Five   hundred    seventy-four    thousand   three    hundred 
thirty-five. 

73.  Seven    hundred    fifty-three   thousand   seven   hundred 
fifty. 

74.  Five  thousand  four  hundred  eighty-nine. 

75.  Four  hundred  eighty  thousand  eight. 

76.  One  million  one  hundred  thirteen. 

77.  Three  million  three  thousand  thirty. 

78.  Nine  hundred  seven  million  eight  hundred  five  thou- 
sand seventy-four. 

79.  Fifty-seven  billion  forty-four  million  ninety-three  thou- 
sand eighty-three. 

80.  Thirteen  million   six  hundred  thousand  one  hundred 
seventy. 

81.  Five   hundred   six   billion  two   million  four  thousand 
one. 

82.  One  thousand  seven  hundred  sixty-two,  and  five  tenths, 

83.  Nine  hundred  ninety-four,  and  thirty-eight  hundredths. 


NOTATION   AND   NUMERATION.  9 

84.  One  hundred  nineteen  thousand,  and  one  hundred  nine- 
teen thousandths. 

85.  One  million  four  hundred  thousand  three  hundred,  and 
thirty-three  hundredths. 

86.  Sixteen  thousand  seven  hundred  forty-five,  and  eighty- 
six  thousandths. 

87.  Three  billions  three  millions  two  thousands  seven  hun- 
•  dreds,  and  three  thousandths. 

EXERCISES    IN    READING    NUMBERS. 

88.  Write  and  r.  ad  25362795. 

Solution.  —  25362795  marked  off  into  groups  be- 
25,362,795      comes  25,362,795.     The  first  group  from  the  right 
expresses  795  ones,  or  795;  the  next  group  expresses 
362  thousands  ;  and  the  third  expresses  twenty-five  millions.     The 
whole  is  read  :    twenty-five  million  three  hundred  sixty-two  thou- 
sand seven  hundred  ninety-five. 

89.  Write  and  read  3257893. 

90.  Write  and  read  64058.109. 

Solution.  —  64058.109  marked  off  into  groups  be- 
64,058  109       comes  64,058.109,  which  is  read  sixty-four  thousand 
fifty-eight,  and  one  hundred  nine  thousandths. 

91.  Write  and  read  1673405.75. 

92.  Write  and  read  906005.805. 

27.    Rule  for  Reading  Numbers. 

Beginning  at  the  ones'  place,  mark  off  the  numbers  by  com- 
mas, into  as  many  groups  as  possible  of  three  figures  each. 

Begin  at  the  left  and  read  each  group  as  if  it  stood  alone, 
giving  the  group-name  except  to  the  ones'  group. 

If  there  is  a  decimal,  read  it  as  if  it  were  at  the  left  of  the 
'point,  and  add  the  place-name  of  the  last  figure. 


10 


NOTATION   AND   NUMERATION. 


Copy  and  read  : 

93.    486 

101. 

6082 

109. 

0.31 

94.   2385 

102. 

19009 

110. 

117.03 

95.   7275 

103. 

163404 

111. 

.064 

96.  12361 

104. 

789685 

112. 

118646.5 

97.  62004 

105. 

99999 

113. 

2567.02 

9a    199 

106. 

1634562 

114. 

88.999 

99.  78382 

107. 

25401300 

115. 

1158834.55 

100.  160405 

108. 

312407981 

116. 

100600.789 

QUESTIONS.  N 

Art.  1.  What  is  a  unit  ?  2.  What  is  a  number  1  3.  What  is 
arithmetic  1 

4.  What  is  notation  ?  5.  What  is  numeration  ?  6.  What  are 
figures  ?  7.  What  does  zero,  or  cipher,  express  ?  What  do  the  other 
nine  figures  express  ? 

9.  When  has  a  figure  a  simple  name  and  value  ?  When  has  it  a 
place-name  and  value  1 

10.  What  is  the  place  of  a  figure?  11.  If  in  the  first  place, what 
place-name  has  a  figure?  What  units  does  it  express?  If  in  the 
second  place,  what  place-name  has  a  figure  ?  What  units  does  it 
express  ? 

12.  What  place-name  has  a  figure  at  the  left  of  tens  ?  What  units 
does  it  express  ? 

13.  What  place-name  has  a  figure  at  the  left  of  hundreds  ?  What 
units  does  it  express  ? 

15.  Give  the  place-names  beginning  with  ones.  Give  the  corre- 
sponding orders  of  units. 

16.  What  form  a  group  for  convenience  in  reading  and  writing 
numbers  ?  How  is  each  group  named  ?  Name  groups  beginning 
with  ones.  What  is  used  to  mark  off"  the  groups  ?  17.  What  does 
each  complete  group  contain  ? 

18.  What  are  principles  of  notation  ?  19.  What  is  our  system 
called  ?    What  do  the  successive  orders  of  units  form  ? 

20.  What  is  the  decimal  point  ?  For  what  is  it  written  ?  21.  What 
is  the  number  at  the  left  of  the  point  ? 

20.  How  arc  numl)er8  written  in  figures?  27.  How  are  numbers 
written  in  fi^cures  read  ? 


ADDITION.  11 


ADDITION. 

28.  1.  James  has  9  apples  and  Henry  has  3.  How  many 
apples  have  both  ? 

2.  Mary  had  7  oranges  and  her  brother  gave  her  5.  How 
many  had  she  then  ? 

3.  A  father  gave  to  one  of  his  children  8  cents,  to  another 
5  cents,  and  to  a  third  6  cents.  How  many  cents  did  he  give 
them  in  all  ? 

4.  How  many  are  9  and  3  ?     7  and  5  ?     S,  5j  and  6  ? 

5.  How  many  dollars  are  8  dollars,  7  dollars,  and  3  dol- 
lars ? 

6    What  is  the  unit  of  8  dollars,  7  dollars,  and  3  dollars  ? 

29.  The  Unit  of  a  number  is  one  of  that  number. 
Thus, 

One  pound  is  the  unit  of  8  pounds,  one  quart  is  the 
unit  of  27  quarts. 

30.  Like   numbers  are  numbers  having  the  same  unit. 
Thus  3,  5,  7 ;  and  6  cents,  4  cents  and  5  cents,  are  like 

numbers. 

31.  Addition  is  finding  a  number  equal  to  two  or  more 
given  numbers. 

32.  The  sum,  or  amount,  is  the  result  of  an  addition. 

33.  The  sign  of  addition  is  +,  named  plus.  It  means 
more,  and  is  generally  read  and.     Thus, 

4  +  5  +  6  is  read  four  and  five  and  six. 

34.  The  sign  of  equality  is  = .  It  means  equal,  or 
equal  to,  and  is  often  read  are.     Thus, 

7  +  8  =  15  is  read  seven  and  eight  are  fifteen. 


12  ADDITION. 

35.  The  sign,  $,  written  before  a  number,  means  dollars. 
Thus, 

$10  is  read  ten  dollars. 

36.  CentSy  expressed  in  figures,  may  occupy  two  deci- 
mal places,  tenths  and  huThdredths,  and  mills  one  place, 
tJwusandths.     Thus, 

$0.53,  or  $.53,  is  read  fifty-three  cents,  and  $28,005  is 
read  twenty-eight  dollars  and  five  mills. 

ORAL     EXERCISES. 

7.  How  many  dollars  are  $  7,  $  6,  and  $  2  ? 

8.  If  you  pay  10  cents  for  a  slate,  9  cents  for  paper,  and 
3  cents  for  a  pencil,  how  much  do  you  pay  for  the  whole  ? 

9.  12  boys  are  at  play  in  one  place,  6  in  another,  and  5  in 
another.     How  many  boys  are  at  plaj^  in  all  ? 

10.  John  one  day  caught  8  trout,  another  day  9,  and  a  third 
day  4.     How  many  did  he  catch  in  the  three  days  ? 

How  many  are : 

11.  5  +  3  17.  2  +  3  +  7  23.  8  +  3  +  7 

12.  8-1-5  18.  1  +  5  +  3  24.  9  -h  1  +  8 
"13.       7  -h  2  19.  3  +  5  -I-  7  25.  7  +  6  +  5 

14.  6  -h  6  20.     5  -f-  2  -f-  6  26.    8  +  2  +  3 

15.  9  +  8  21.     4  +  1  +  8  27.    3  +  6  +  7 

16.  11  +  7  22.     6  +  0  +  9  28.    9  +  7  +  8 

29.  Add  by  2's  from  0  to  24,  naming  only  results. 
Solution.  —  0,  2,  4,  6,  8,  10,  12,  14,  16,  18,  20,  22,  24. 

Add: 

30.  By  2's  from  1  to  25.  36.  By  4's  from  2  to  34. 

31.  By  3's  from  2  to  29.  37.  By  4's  from  3  to  35. 

32.  By  3's  from  3  to  30.  38.  By  5's  from  1  to  31. 

33.  By  3'8  from  4  to  31.  39.  By  5's  from  2  to  32. 

34.  By  4's  from  0  to  32.  40.  By  5's  from  3  to  33. 

35.  By  4'8  from  1  to  33.  41.  By  5's  from  4  to  34. 


ADDITION.  13 

42.  How  many  are  8  +  6  +  5?  5  +  6  +  8?  8  +  5  +  6? 
6+8+5? 

37.     Principles  of  Addition. 

1.  Only  like  numbers^  and  units  of  the  same  order  can  he 
added. 

2.  The  sum  is  the  same  in  whatever  order  the  numbers 
are  added. 

WRITTEN     EXERCISES. 

43.  What  is  the  sum  of  142,  16,  201,  and  410  ? 

^  .^  Solution.  — We  write  the  numbers  so  that  units 

of  the  same  order  may  he  in  the  same  column. 
Beginning  with  ones,  we  add,  naming  results, 
^^^  thus  :   0,  1,  7,  9,  and  writing  the  9  beneath  in 

410  ones'  place. 

769  Adding  the  tens,  1,  0,  2,  6,  the  sum,  6  tens, 

we  write  beneath  in  tens'  place. 
Adding  the  hundreds,  4,  6,  7,  the  sum,  7  hundreds,  we  write  be- 
neath in  hundreds'  place. 

The  sum,  then,  is  7  hundreds  6  tens  9  ones,  or  769. 
To  test  or  prove  the  work  we  add  the  columns  downward,  and 
have,  as  before,  the  sum  769. 

44.  What  is  the  sum  of  121,  516,  361,  ftnd  11  ? 

45.  What  is  the  sum  of  231  4-  114  +  324  ? 

46.  How  many  are  235  +  321  +  142  ? 

47.  Sold  416  bushels  of  corn  to  one  man,  301  to  another, 
and  42  to  a  third.     How  many  bushels  were  sold  in  all  ? 

48.  Paid  L  r  labor  $104,  for  boards  $530,  for  timber  $243, 
<»,nd  for  hardware  $112.     How  much  was  paid  for  all  ? 

49.  Bought  a  horse  for  $150,  a  carriage  for  $200,  a  harness 
for  $45,  and  hay  and  grain  for  $104.  What  was  the  cost  of 
the  whole  ? 

50.  Mr.  Smith,  in  his  will,  gave  to  his  son  Arthur  $500 ;  to 
his  son  John  $365;  to  his  daughters  $475,  and  to  his  brother 
$  125.     How  much  did  he  give  them  ill  ? 


14  ADDITION. 


ORAL    EXERCISES. 


51.  In  one  basket  there  are  20  apples,  in  another  9,  and  in 
a  third  7.     How  many  are  there  in  all  ? 

Add:  • 

52.  By  5's  from  5  to  35.  58.  By  7's  from  1  to  43. 

53.  By  6's  from  0  to  36.  59.  By  7's  from  2  to  44. 

54.  By  6's  from  1  to  37.  60.  By  8's  from  3  to  43. 

55.  By  &s  from  3  to  39.  61.  By  8's  from  4  to  44. 

56.  By  6's  from  4  to  40.  62.  By  9's  from  1  to  46. 

57.  By  6's  from  5  to  41.  63.  By  9's  from  2  to  47. 

64.  Add  by  7's  from  4  to  39;  from  5  to  40  ;  from  6  to  41. 

65.  Add  by  8's  from  1  to  41 ;  from  2  to  42  ;  from  5  to  45; 
from  6  to  46  ;  from  7  to  47. 

66.  Add  by  9's  from  3  to  48  ;  from  4  to  49 ;  from  5  to  50; 
from  6  to  51 ;  from  7  to  52. 

67.  Add  by  7's  from  0  to  42  ;  by  8's  from  0  to  48  ;  by  9's 
from  0  to  54. 

How  many  are : 

68.  17  +  2  +  5  +  6  ?  70.   13  +  2  +  7  +  4  ? 

69.  21  +  5  +  8  +  2  ?  71.  32  +  6  +  3  +  9  ? 

72.  What  is  the  sum  of  67  and  25  ? 

73.  If  you  should  travel  one  day  20  miles  and  the  next  day 
24  miles,  how  far  would  you  travel  in  the  two  days  ? 

74.  A  man  earned  in  one  week  $11,  in  another  $16,  and  in 
a  third  $13.     How  much  did  he  earn  in  all  ? 

75.  A  father  gave  to  one  of  his  sons  30  cents,  to  another 
14  cents,  and  to  his  daughter  12  cents.  How  much  did  he 
give  them  in  all  ? 

76.  In  one  iield  there  are  55  acres,  in  another  40,  and  in 
the  third  17.     How  many  acres  are  there  in  the  three  fields  ? 


ADDITION.  15 

77.  Johnson  keeps  on  one  farm  60  cows,  on  another  19, 
and  on  a  third  21.     How  many  does  he  keep  on  all  ? 

WRITTEN     EXERCISES. 

78.  What  is  the  sum  of  595,  961,  and  23. 

595  Solution.  —  We  write  the  numbers  so  that  units 

961  of  the  same  order  may  be  in  the  same  column. 

^^  Beginning  with  ones,  we  add,  thus  :    3,  4,  9  ; 

1579  sum,  9  ones,  which  we  write  beneath  in  ones* 

place. 
We  add  the  tens,  thus:  2,  8,  17;  sum,  17  tens,  which  are  170,  or  1 
hundred  7  tens.     We  write  the  7  tens  beneath  in  tens'  place,  and  add 
the  1  hundred  with  the  hundreds  in  the  next  column. 

We  add  the  hundreds,  thus:  1,  10,  15  ;  sum,  15  hundreds,  which  are 
1500,  or  1  thousand  5  hundreds.     We  write  the  5  hundreds  beneath 
in  hundreds'  place,  and  the  1  thousand  in  thousands'  place. 
The  sum  is  1  thousand  5  hundred  7  tens  9  ones,  or  1579. 
To  prove  the  work,  we  add  the  columns  downward,  and  have,  as 
before, the  sum  1579. 

79.  Add  626,  317,  529,  and  12. 

80.  Add  368,  689,  73,  and  19. 

81.  Add  $  21.642,  $  0.763,  $  3.05,  and  $  5.90. 

$21,642  Solution.  —  Beginning  with    thousandths,   we 

0.763  ^^^>  thus:  3,  6;  sum,  5  thousandths,  which  we 

2  QK  write  beneath  in  thousandths'  place. 

K  qrw  We  add  the  hundredths,  thus:  5,  11,  15;  sum, 

15  hundredths,  or  1  tenth  5  hundredths.  We  write 

$  ol.ooo  ^jjQ  5  hundredths  beneath  in  hundredths'  place, 

and  add  the  1  tenth  with  the  tenths  of  the  next  column. 

We  add  the  tenths,  thus:  1,  10,  17,  23;  sum,  23  tenths,  or  2  ones 
3  tenths.  We  write  the  3  tenths  beneath  in  the  place  of  tenths,  and 
add  the  2  ones  with  the  ones  of  the  next  column. 

We  add  the  ones,  thus:  2,  7,  10, 11;  the  sum,  11  ones,  or  1  ten  1  one. 
We  write  the  1  one  beneath  in  the  ones'  place,  and  add  the  1  ten  with 
the  tens  of  the  next  column. 

We  add  the  tens,  thus:  1,  3;  sum, 3  tens,  which  we  write  beneath 
in  tens'  place. 

The  sum  is  |  31.355,  or  31  dollars  and  35  cents  5  mills. 


16  ADDITION. 

82.  What  is  the  sum  of  $31,067,  $8,091,  $0.46,  and 
$  0.31  ? 

83.  What  is  the  sum  of  $13,615,  $24.25,  $6.14,  and 
$  17.66  ? 

84.  What  is  the  sum  of  $91.55,  $82.35,  $63,  and 
$  80.50  ? 

38.     Rule  for  Addition. 

Write  the  numbers  so  that  units  of  the  same  order  may 
be  in  the  same  column. 

Add  the  right-hand  column,  writing  the  units  of  the  sum 
beneath,  and  adding  the  tens,  if  any,  to  the  next  column. 

So  2)roceed  with  all  the  columiis,  writing  the  entire  sum  of 
the  last  column. 

39.  Proof.  — Add  the  numbers  a  second  time  and  in 
a  different  order  (Art.  37). 


85. 

86. 

87. 

88. 

346 

800 

1123 

5050 

275 

455 

678 

7825 

62 

619 

642 

1367 

6 

104 

53 

1243 

89. 

90. 

91. 

92. 

315 

6541 

841.5 

1.3.14 

640 

1809 

302.6 

21.65 

179 

723 

427.5 

82.91 

106 

480 

122.4 

11.78 

812 

235 

324.7 

14.43 

93. 

94. 

95. 

96. 

%  53.19 

$93.47 

%  103.64 

*  111.625 

61.25 

6.80 

205.75 

41.55 

16.87 

0.39 

83.91 

76.875 

X* 


ADDITION.  17 

97.  Find  the  sum  of  406,  781,  918,  846,  and  67. 

.     98.  Find  the  sum  of  1081,  686,  423,  925,  and  328. 

99.  Find  the  sum  of  2642,  3417,  506,  689,  and  482. 

100.  Find  the  sum  of  391,  638,  402,  83,  493,  and  16. 

101.  Find  the  sum  of  555,  64,  380,  75,  878,  and  74. 

102.  Find  the  sum  of  159,  363,  4682,  8405,  and  3377 

How  many  are : 

103.  62.48  +  31.41  +  8.95  +  73  +  0.56  ? 

104.  8.15  +  31.68  +  9.60  +  18.53  +  85.93  ? 

105.  106  +  57.25  +  46.17  +  9.36  +  62.94  ? 

106.  3.19  +  11.93  +  61.85  +  376  +  4781  ? 

107.  $  16.43  +  $  24.77  +  $  75.35  +  $  9.10  ? 

108.  $  342  +  $  164  +  $  4.95  +  $  0.74  ? 

109.  $  78.05  +  $  72.09  +  $  8.11  +  $  9.83  ? 

110.  $415  +  $88.24  +  $13.08  +  $16.08? 

111.  On  a  farm  105  trees  bear  pears,  492  bear  peaches,  85 
bear  cherries,  and  316  bear  apples.  How  many  trees  are  there 
in  all  ? 

112.  An  army  consisted  of  2358  infantry,  868  cavalry,  and 
1165  artillery.     What  was  the  number  of  the  army  ? 

113.  A  man  bought  a  house  for  $  8750.  He  paid  $36355 
lov  repairs,  $95.63  for  painting,  and  $  106.50  for  taxes.  For 
how  much  must  he  sell  it  to  gain  $  350  ? 

114.  A  farmer  has  four  fat  oxen.  The  first  weighs  1463 
pounds,  the  second  1385  pounds,  the  third  1507  pounds,  and 
the  fourth  1264  pounds.     What  is  the  weight  of  them  all  ? 

115.  Bought  a  suit  of  clothes  for  $42.50,  an  overcoat  for 
$  22.75,  a  pair  of  boots  for  $  5.25,  and  a  hat  for  $  4.63.  What 
was  the  cost  of  the  whole  ? 

116.  The  area  of  France  is  204091  square  miles  and  that  of 
Italy  114290.      What  is  the  area  of  the  two  countries  ? 

117.  What  is  the  sum  of  $103,  $0.06,  $15.05,  $19.75,  and 
$7.31? 

2 


18  ADDITION. 

118.  Gave  $  73  for  a  watch,  $  15.50  for  a  carriage  robe. 
$  250  for  a  horse,  and  sold  them  so  that  I  gained  $  21.50. 
What  did  I  sell  them  for  ? 

119.  A  merchant  began  business  with  goods  worth  $  6750  ; 
a  store  worth  $  5700 ;  fixtures  worth  $  555.25  j  and  gained 
1 1165.45.     What  was  he  then  worth  ? 


120. 

1,21. 

122. 

123. 

321 

8106 

31.46 

93.045 

406 

7334 

42.47 

10.304 

718 

5570 

60.05 

7.105 

304 

2344 

63.06 

16.321 

818 

648 

71.85 

4.554 

103 

102 

40.25 

17.056 

145 

341 

14.40 

2.005 

124.  Mount  Everest  is  13270  feet  higher  than  Mont  Blanc, 
Mont  Blanc  4832  feet  higher  than  Etna,  and  Mount  Etna  rises 
10900  feet  above  the  level  of  the  sea.  What  is  the  height  of 
Mount  Everest  above  the  sea  level  ? 

125.  The  area  of  the  United  States  consists  of  territory 
ceded  by  Great  Britain  as  the  result  of  the  Revolution, 
815615  square  miles ;  acquired  from  France,  930928  square 
miles  ;  from  Spain,  59268  square  miles;  by  admission  of  Texas, 
237504  square  miles;  Oregon  by  treaty,  280425  square  miles  ;- 
from  Mexico,  677262  s<|uare  miles ;  and  from  Russia,  577390 
square  miles.     What  u  the  total  area  of  the  United  States  ? 

126.  In  building  a  cottage,  the  excavating  cost  $  34 ;  the 
cellar  walls,  $  110,50 ;  chimneys,  $  18.46 ;  the  plastering, 
$73.42;  the  frame,  $64.50;  the  boarding,  $33.50;  the  sid- 
ing, «5  25;  tli^  roof  boards,  $20.67;  the  shingling,  $62.80; 
the  gutters  and  hardware,  $  42 ;  ti)e  truss  work,  $13;  the 
water  tank,  $  15  ;  the  cornices,  $  46  ;  the  windows,  $  98  ;  the 
doors,  $126;  tlie  flooring,  $45;  the  stairs,  $20;  the  base, 
*34.40;  the  sink,  $9.50;  the  cistern,  $25;  the  painting, 
$80.12;  and  incidentals,  $50.     What  was  the  entire  cost? 


ADDITION.  19 

X^127.  Bought  a  house  for  $23650  and  land  for  $73640. 
Paid  $  4500  for  repairs  and  taxes.  For  house  and  land  how- 
much  must  I  receive  to  gain  the' cost  of  the  land  ? 

128.  The  first  of  four  numbers  is  8437,  the  second  9325, 
the  third  is  the  sum  of  the  first  two,  and  the  fourth  is  the  sum 
of  the  second  and  third.  Find  the  result  if  the  numbers  are 
T)ut  together. 

129.  A,  B,  C,  and  D  go  into  business.  A  puts  in  $7430, 
B  $  3200  more,  C  as  much  as  A  and  B  together,  and  D  as 
much  as  A  and  C  together.     What  is  the  capital  of  the  firm  ? 

130.  In  1880  the  Boston  and  Providence  R.  B.  received 
from  passengers  $  776362.87,  from  freight  $486724.85,  from 
rents  $  19395.08,  from  express  companies  $  30202.34,  and  from 
mails  $  11240.49.     What  was  the  total  income  for  the  year  ? 

131.  Find  the  sum  of  the  numbers  in  Art.  26,  Exercises  66 
to  74  inclusive. 

132.  In  Exercises  73  to  81  inclusive. 

133.  In  Exercises  81  to  87  inclusive. 

134.  In  Art.  27,  Exercises  93  to  100  inclusive. 

135.  In  Exercises  101  to  108  inclusive. 

136.  In  Exercises  109  to  116  inclusive. 

137.  What  are  my  sales  for  the  week  if  my  daily  sales  are 
as  follows:  Monday  $347.19,  Tuesday  $847.62,  Wednesday 
$9643.27,  Thursday  $9876.50,  Friday  $843.91,  Saturday 
$10986.75? 

138.  What  is  the  income  of  a  gentleman  who  receives 
annually  from  rents  $  2465.29,  from  mining  profits  $  3462. 
from  other  business  $9478.50,  and  from  interest  of  U.  S.  Bonds 
$8600? 

QUESTIONS. 

29.  What  is  the  unit  of  a  number  ?  30.  What  are  like  numbers  ? 
31.    What  is  addition  ?     32.   What  is  the  sum  ? 

33.  What  is  the  sign  of  addition  ?  34.  The  sign  of  equality  1 
35.    The  dollar  sign? 

37.  What  are  principles  of  addition  ?  How  do  you  add  ?  What 
ts  the  proof  ? 


20  SUBTRACTION. 


SUBTRACTION. 

40.  1.   John  has  8  marbles,  and  his  brother  5.    How  many 
more  marbles  has  John  than  his  brother  ? 

2.  Thomas  had  9  apples,  and  gave  4   of   them  to   Peter 
How  many  had  he  left  ? 

3.  How  much  more  are  11  cents  than  6  cents  ? 

4.  How  many  dollars  are  $  13  less  $  7  ? 

5.  Ella  is  14   years  old,  and  Mary  is  9  years  old.     How 
many  years  older  is  Ella  than  Mary  ? 

6.  Sold  a  knife  for  15  cents  and  a  ball  for  7  cents.     How 
much  more  did  I  get  for  the  knife  than  for  the  ball  ? 

41.  Subtraction  is  taking  one  of  two  like  numbers  from 
the  other. 

42.  The  Difference  is  the  result  of  a  subtraction. 

43.  The  Minuend  is  the  number  subtracted  from. 

44.  The  Subtrahend  is  the  number  subtracted. 

45.  The  Sign  pf   Subtraction  is  —  ,  named  minus.     It 
means  less.     Thus, 

15  —  9  =  6  is  read  fifteen  less  nine  are  six. 

ORAL     EXERCISES. 

7.  A  boy  raised  13  melons  and  sold  6.     How  many  had  he 
left? 

8.  Arthur  had  11  cents  and  gave  away  7.     How  many  had 
he  left? 

9.  In  a  nest  there  were  14  eggs,  but  5  have  been  taken 
away.     How  many  remain  in  the  nest  ? 

10.  In  a  pool  were  13  lilies,  and  8  have  been  carried  away. 
How  many  remain  ? 

11.  If  of  16  peaches  8  should  be  eaten,  how  many  would  be 
I(;ft  ? 


SUBTRACTION.  21 

How  many  are : 

12.  16-9         17.  27->    9  22.  18-9  26.  21-5 

13.  15-7         18.  19-10  23.  25-6  27  24-6 

14.  11-5         19.  26-8  24.  19-9  28.  20-7 

15.  13-8         20.  17-3  25.  23-8  29.  28-9 

16.  14-6         21.  12—7 

30.  Subtract  by  2's  from  40  back  to  20,  naming  only  re- 
sults. 

31.  By  3's  from  38  to  24.  37.    By  5's  from  58  to  28. 

32.  By  4's  from  37  to  21.  38.    By  4's  from  56  to  40. 

33.  By  5's  from  51  to  26,  39.   By  7's  from  57  to  36. 

34.  By  6's  from  49  to  31.  40.   By  6's  from  54  to  30. 

35.  By  7's  from  50  to  29.  41.    By  8's  from  53  to  31. 

36.  By  8's  from  48  to  32.  42.   By  9's  from  49  to  22. 

43.  How  many  are  $  16  less  $  9  ?     $  7  and  how  many  dol^ 
lars  are  $  16  ? 

46.     Principles  of  Subtraction. 

1.  Only  like  numbers  and  units  of  the  same  order  can  he 
subtracted  one  from  the  other, 

2.  The  difference  and  subtrahend  together  must  equal  the 
minuend. 

WRITTEN    EXERCISES. 

44.  Find  the  difference  between  865  and  242. 

Minuend  865  Solution.  —  For    convenience,    we 

Subtrahend     242  ^^^^^  *^^  subtrahend  under  the  min- 

uend,  so  that  units  of  the  same  order 

Difference       623  ^^^  ^^  ^^  ^^^  ^^^^^  ^^1^,^^^^ 

Proof  865  2  ones  from  5  ones  leave  3  ones, 

which   we   write    beneath   in   ones' 

place ;  4  tens  from  6  tens  leave  2  tens,  which  we  write  beneath  in 

tens'  place;   and  2  hundreds  from  8  hundreds   leave  6   hundreds, 

which  we  write  beneath  in  hundreds'  place. 

The  difference  is  6  hundreds  2  tens  3  ones,  or  623. 


SUBTRACTION.  23 

73.  John  travels  35  miles  a  day  and  Edwin  60  miles.  How 
many  more  miles  does  the  one  travel  than  the  other  ? 

74.  A  boy  had  51  cents  and  spent  38.  How  many  cents 
had  he  left  ? 

WRITTEN     EXERCISES. 

75.  What  is  the  difference  between  1624  and  342? 

Minuend          1624  Solution.  —  2  ones  from  4  ones 

Subtrahend       342  ^^^^^  ^  ones,  which  we  write  be^ 

neath  in  the  place  of  ones. 

Difference       1282  ^  ^^^^  ^^^^^^^  ^^  ^^^^^  ^^^^  2 

Proof  1624  tens  ;  we  therefore  take  1  hundred, 

or  10  tens,  from  the  6  hundreds  of 

the  minuend,  leaving  5  hundreds,  and  adding  the  10  tens  to  the  2 

tens  we  have  12  tens  ;  4  tens  from  12  tens  leave  8  tens,  which  we 

write  beneath  in  the  place  of  tens. 

3  hundreds  from  5  hundreds  leave  2  hundreds,  which  we  write  in 
hundreds'  place  ;  no  thousands  from  1  thousand  leaves  1  thousand, 
which  we  write  in  thousands'  place. 

The  difference  is  1282.  This  we  prove  by  adding  the  difference 
and  subtrahend,  and  finding  the  sum  to  be  equal  to  the  minuend. 

76.  Find  the  difference  between  167  and  476. 

77.  Find  the  difference  between  389  and  581. 

78.  What  number  and  1543  make  1735  ? 

79.  Subtract  32.15  from  78.8. 

Minuend         78.80  Solution.  —  We  make  the  deci- 

Subtrahend     32.15  ^^1  places  the   same  in  the  two 

numbers   by   filling  the  place   of 

Difference       46.65  hundredths  in  the  minuend  by  0. 

5  hundredths  cannot  be  taken 
from  0  hundredths;  we  therefore  take  1  tenth, or  10  hundredths,  from 
the  8  tenths,  leaving  7  tenths;  5  hundredths  from  the  10  hundredths 
leave  5  hundredths,  which  we  write  beneath  in  hundredths'  place. 

1  tenth  from  7  tenths  leaves  6  tenths,  which  we  write  beneath  in 
tenths'  place  ;  and  subtracting  the  ones  and  tens,  and  writing  the  re- 
sult beneath,  we  have  46.65  as  the  result  required. 


24  SUBTRACTION. 

80.  Subtract  167.807  from  970.96. 

81.  Take  $158.55  from  $549.60. 

82.  How  much  more  than  $91.97  is  $147.11  ? 

83.  How  much  and  $  6711.45  make  $  7050.25  ? 

47.     Rule  for  Subtraction, 

Write  the  subtrahend  under  the  minuendj  placing  units  of 
the  same  order  in  the  same  column. 

Begin  with  the  units  of  the  lowest  order  to  subtract,  and 
proceed  to  the  highest,  W7'iting  the  result  beneath. 

If  any  order  of  the  minuend  has  less  units  than  the  same 
order  of  the  subtrahend,  increase  its  units  by  ten,  and  subtract ; 
consider  the  units  of  the  next  minuend  order  one  less,  and  pro- 
ceed as  before. 

48.  Proof.  Add  the  subtrahend  and  the  difference 
together ;  the  sum  should  equal  the  minuend  (Art.  46). 

84.  85.  86.  87. 

From  8647       5375       6365       9406 
Take  3451       _406       4506       8350 

88.  89.  90.         91. 

From    31867  12805  148.48  63.859 

Take      1905  9264  92.09  49.608 

92.  93.  94.  95. 

From     $63.95  $85.69  $96.70  $182.05 

57.68  1.73  0.89  152.06 

96.   Find  the  difference  between  34000  and  21345. 

(8X9X9X10)  Solution.  —  There  being  0  ones,  0  tens,  0 

3  4  0  0  0  hundreds  in  the  minuend,  we  take  1  of  the  4 

213  4  5  thousands  (leaving  3  thousands),  or  10  hun- 

dreds;  then  taking  1  of  the  10  hundreds  (leav- 

12  6  5  5  iiig  9  hundreds), or  10  tens ;  and  1  of  the  10 

tens  (leaving  9  tens),  or  10  ones;  the  minuend 


SUBTRACTION.  25 

may  then  be  considered  3  ten-thousands  3  thousands  9  hundreds 
9  tens  and  10  ones. 

Taking  from  the  changed  minuend  the  2  ten-thousands  1  thousand 
3  hundreds  4  tens  5  ones  of  the  subtrahend,  we  have  as  the  difference 
required,  12655. 

Subtract : 

97.  5544  from  40000.  103.  67.055  from  73.607. 

98.  180  from  98000.  104.  19.55  from  831.50. 

99.  12453  from  35421.  105.  444.5  from  10060. 

100.  97  from  10000.  106.    921.56  from  1000. 

101.  58346  from  67500.  107.   $13.63  from  $500.20. 

102.  9999  from  10000.  108.    $  127  from  $  1963.75. 

109.  What  is  the  value  of  83956  -  78415  ? 

110.  What  is  the  value  of  60440  -  33457  ? 

111.  What  is  the  value  of  109800  -  98799  ? 

112.  From  one  hundred  nine  take  one,  and  nine  hun- 
dredths ? 

113.  America  was  discovered  in  1492 ;  how  many  years  from 
that  date  to  1881  ? 

114.  A  man  began  business  with  $1760,  and  after  two 
years  had  $  2500.75.     How  much  had  he  gained  ? 

115.  The  number  of  regular  soldiers  furnished  by  the  sev- 
eral States  in  the  war  of  the  Revolution  was  231771 ;  of  these 
a  single  State  furnished  67907.  How  many  were  furnished 
by  other  States  ? 

116.  A  merchant  bought  goods  to  the  amount  of  $  7563.56;, 
and  sold  them  for  1 11630.50.     How  much  did  he  gain  ? 

117.  The  Atlantic  slope  contains  967576  square  miles,  and 
the  Mississippi  valley  1237111  square  miles.  How  much  does 
the  latter  exceed  the  former  ? 

118.  The  population  of  Kew  York  City  in  1870  was  942192, 
and  in  1880  was  1209561.     How  much  was  the  gain  ? 

119.  The  product  of  gold  in  a  certain  year  was  from  Nevada 
$19546516,  and  from  California  $17760679.  How  much 
greater  was  the  product  from  Nevada  than  from  California  ? 


26  SUBTRACTION. 

MISCELLANEOUS     EXERCISES. 

120.  A  man  owing  $  767.50  paid  one  time  $  190,  at  another 
8  131,  and  at  a  third  $  155.25.     How  much  did  he  then  owe  ? 

121.  James  Dow's  real  estate  is  valued  at  $  3769,  and  his 
personal  estate  at  $2648.75.  He  owes  Job  Smith  $1728 
and  Abraham  Tyler  $  1161.93.  How  much  is  he  worth  wher 
these  debts  are  paid  ? 

122.  Modern  notation  in  arithmetic  was  introduced  from 
Arabia  into  Europe  in  the  year  991,  algebra  from  the  same 
country  in  431  years  later,  and  decimal  fractions  were  invented 
in  1602.     Required  the  number  of  years  from  each  to  1882. 

123.  Sydney  has  $  178.50,  Albert  $  75.75  more  than  Sydney, 
and  Charles  has  as  much  as  Sydney  and  Albert  less  $  80.93. 
How  much  more  has  Charles  than  Sydney  ? 

124.  What  is  the  value  of  1645  +  635  +  416  -  1314  ? 

125.  Two  vessels  4563  miles  apart  start  to  meet  each  other. 
When  one  of  them  has  made  1575  miles  of  the  distance  and 
the  other  1658  miles,  how  far  are  they  apart  ? 

126.  A  man  whose  property  was  $  50675  gave  his  son 
Edwin  $  8555.50,  his  son  Eobert  $7000,  his  daughter  Mary 
$  9563.75,  his  wife  $  20000,  and  a  public  library  the  remainder. 
How  much  was  given  the  public  library  ? 

127.  The  population,  in  1880,  of  Philadelphia  was  847542 ; 
of  Boston,  362535;  and  of  Providence,  \04760.  How  much 
greater  was  the  population  of  Philadelphia  than  that  of  Boston 
and  Providence  ? 

128.  A  man  who  had  $  1250  in  a  savings  bank  took  out 
$  51.75  at  one  time,  $  84.93  at  another,  and  $  267  at  another. 
He  then  put  in  $  185.     IJow  much  then  had  he  in  the  bank  ? 

QUESTIONS. 

41.  What  is  Bubtraction  ?  42.  What  is  the  difference  ?  43.  The 
minuend?    44.   The  subtrahend  ?     45.    The  sign  of  subtraction  ? 

46.  What  are  principles  of  subtraction  ?  47.  How  do  you  sub- 
tract ?  If  any  order  of  the  minuend  has  less  units  than  the  same 
order  of  the  subtrahend,  what  is  done  ?  48.  What  is  the  proof  of 
Bubtraction? 


MULTIPLICATION.  27 


MULTIPLICATION. 

49.  1.  If  a  man  can  earn  8  dollars  in  a  week,  how  many 
dollars  can  he  earn  in  4  weeks  ?  How  many  dollars  are  4 
times  8  dollars  ? 

2.  What  will  5  quarts  of  cherries  cost  at  10  cents  a  quart  ? 

3.  William  has  9  apples  and  Alfred  7  times  as  many.  How 
many  has  Alfred  ?     7  times  9  are  how  many  ones  ? 

50.  Multiplication  is  taking  one  number  as  many  times 
as  there  are  ones  in  another. 

51.  The  Multiplicand  is  the  number  taken  or  multi- 
If^lied. 

52.  The  Multiplier  is  the  number  that  shows  how  many 
times  the  multiplicand  is  taken. 

53.  The  Product  is  the  result  of  a  multiplication  ;  and 
the  factors  of  a  product  are  the  numbers  multiplied  to- 
gether to  produce  it. 

54.  The  Sign  of  Multiplication  is  x.  It  means  multi- 
plied by,  or  tiincs.     Thus, 

7x5  may  be  read  seven  multiplied  by  five,  or  seven 
times  five. 

55.  A  Concrete  Number  is  a  number  in  which  some 
kind  of  unit  is  named.     Thus, 

2  books,  3  days,  $  7,  are  concrete  numbers. 

56.  An  Abstract  Number  is  a  number  in  which  no  par- 
ticular kind  of  unit  is  named.     Thus, 

2,  5,  7  are  abstract  numbers. 


28 


57, 


1X1  = 

1 

1X2  = 

2 

1X3  = 

3 

1X4  = 

4 

1X5  = 

5 

1X6  = 

6 

1X7  = 

7 

1X8  = 
1X9  = 

8 
9 

2  X    9=    18 

3X9  = 

27 

4X9  = 

36 

1  X  10  = 

10 

2  X  10  =    20 

3  X  10  = 

30 

4  X  10  = 

40 

1  X  11  = 

11 

2  X  11  =    22 

3X  11  = 

33 

4  X  11  = 

44 

1  X  12  = 

12 

2  X  12=    24 

3  X12  = 

36 

4  X  12  = 

48 

5X1  = 

5 

6X1=      6 

7X    1  = 

7 

8X1  = 

8 

5X2  = 

10 

6  X    2=    12 

7X2  = 

14 

8X2  = 

16 

5X3  = 

15 

6  X    3=    18 

7X3  = 

21 

8X3  = 

24 

5X4  = 

20 

6  X    4=    24 

7X    4  = 

28 

8X4  = 

32 

5X5  = 

25 

6  X    5  =    30 

7X5  = 

35 

8X5  = 

40 

5X6  = 

30 

6  X    6  =    36 

7X6  = 

42 

8X6  = 

48 

5X7  = 

35 

6  X    7  =    42 

7X7  = 

49 

8X7  = 

56 

5X8  = 

40 

6  X    8  =    48  • 

7X8  = 

56 

8X8  = 

64 

5X9  = 

45 

6  X    9  =    54 

7X9  = 

63 

8X9  = 

72 

5  X  10  = 

50 

6  X  10=    60 

7  X  10  = 

70 

8  X  10  = 

80 

5  Xll  = 

55 

6  X  11  =    66 

7  X  11  = 

77 

8  X  11  = 

88 

5  X12  = 

60 

6  X  12=  -72 

7  X  12  = 

84 

8  X  12  = 

96 

9X1  = 

9 

10  X    1  =    10 

11  X    1  = 

11 

12X1  = 

12 

9X2  = 

18 

10  X    2  =    20 

11  X    2  = 

22 

12X    2  = 

24 

9X3  = 

27 

10  X    3  =    30 

11  X    3  = 

33 

12  X    3  = 

36 

9X4  = 

36 

10  X    4  =    40 

11  X    4  = 

44 

12  X    4  = 

48 

9X5  = 

45 

10  X    5=    50 

11  X    5  = 

55 

12  X    5  = 

60 

9X6  = 

54 

10  X    6=    60 

11  X    6  = 

66 

12  X    6  = 

72 

9X7  = 

63 

10  X    7  =    70 

11  X    7  = 

77 

12  X    7  = 

84 

9X8  = 

72 

10  X    8=    80 

11  X    8  = 

88 

12  X    8  = 

96 

9X9  = 

81 

10  X    9=    90 

11  X    9  = 

99 

12  X    9  = 

108 

9X10  = 

90 

10  X  10  =  100 

11  xio  = 

110 

12X  10  = 

120 

9  Xll  = 

99 

10  X  11  =  110 

11X11  = 

121 

12X  11  = 

132 

9  X12  = 

108 

10  X  12  =  120 

11  X  12  = 

132 

12  X  12  = 

144 

MULTIPLICATION.  29 


ORAL    EXERCISES. 


4.  How  many  wings  have  8  doves  ? 

5.  How  much  can  you  earn  in  3  days  if  you  earn  10  cents 
a  day  ? 

6.  At  9  cents  each  what  will  a  half-dozen  writing-books  cost  ? 

7.  How  many  horns  have  8  yoke  of  oxen  ? 

8.  How  many  days  in  12  weeks  ?     In  9  weeks  ? 

9.  How  far  can  you  ride  in  7  hours  at  the  rate  of  8  miles 
an  hour  ? 

10.  How  many  desks  in  a  school-room  having  8  rows  with 
7  desks  in  a  row  ? 

11.  How  many  legs  have  6  span  of  horses  ? 

12.  At  7  cents  each  what  cost  a  dozen  pencils  ? 

13.  A  ten-foot  pole  is  how  many  inches  long  ? 

How  many  are: 


14. 

15. 

16. 

17. 

7X9 

6x12 

2x    7 

6x    7 

8X8 

8x10 

10X12 

12x12 

9x6 

6x31 

8X7 

11  X    8 

7X7 

9x12 

9X    4 

9x    7 

8x3 

8x10 
18. 

11X11 

19. 

4X    7 

12  X 

9 

-8x12 

6x    9- 

7x    7 

6x 

6 

-4x    7 

8x    6- 

4x10 

7x 

8 

-5x    9 

7X12- 

9-    9 

6x 

7 

-6x    6 

12  X  11  - 

10x10 

9X 

0 

X8x    4 

12  X    6- 

8x    8 

20.    Give  the  products  by  2,  from  2  X  2  to  2  X  12. 
Solution.  —  2,  4,  6,  8,  10,  12,  14,  16,  18,  20,  22,  24. 

2L    Give  the  products  by  4,  from  4  X  0  to  4  X  12. 

22.  Count  by  7's  from  0  to  84. 

23.  Subtract  by  9's  from  108  to  0. 


30  MULTIPLICATION. 

24.  Name  all  the  products  of  which  12  is  a  factor  to  144. 

25.  What  cost  9  six-cent  hooks  ?     12  ?     8  ? 

26.  What  will  12  dozen  eggs  cost  at  a  cent  apiece  ? 

27.  How  many  days  in  a  9  weeks'  vacation  ? 

28.  How  many  corners  have  11  squares  ? 

29.  I  pay  7  cents  daily  for  milk.       What  is  my  milk  bill 
for  a  week  ? 

30.  How  many  inches  in  8  feet  ?     In  9  ?     In  11  ? 

31.  How  many  feet  in  12  yards  of  ribbon  ? 

32.  What  cost  7  lbs.  at  5  cents  a  pound,  and  5  quarts  at  4 
cents  a  quart  ? 

33.  What  numbers  multiplied  make  36  ?     56?     81?    96  ? 

34.  What  factors  produce  48  ?     72  ?     121  ?     63  ?     49  ? 

35.  At  $  8  a  barrel,  how  many  times  $  8  will  9  barrels  of 
flour  cost  ? 

36.  How  many  are9x$8?     8x$9?     3x2x2?     2 
K3x2?2X2x3? 

58.     Principles  of  iVIultiplication. 

1.  The  multiplier  is  always  considered  an  abstract  number. 

2.  The  product  and  the  multiplicand  are  always  like  num- 
bers, 

3.  The  product  is  the   same   whatever  the  order  of  the 
factors, 

WRITTEN    EXERCISES. 

37.  Find  the  product  of  274  multiplied  by  7. 

,  Ti*-  1,.  -1.        1  r^^A  Solution.  —  We  write 

T2,    ,            (  Multiplicand  274  ^,          ,,.  ,.       ^ 

Factors      <            ^  the  multiplier,  7   ones, 

(  Multiplier           7  ^^^^^  ^^^  ^^^g,  ^^^^.^  ^^ 

Product  1918  the  multiplicand. 

Beginning  with  the 
onee,  we  multiply  :  7  times  4  ones  are  28  ones,  or  2  tens  8  ones.  We 
write  the  8  ones  beneath  in  the  place  of  ones,  and  reserve  the  2  tens. 
7  times  7  tens  are  49  tens,  and  49  tens  plus  the  2  tens  reserved  are 
51  tens,  or  5  hundreds  1  ten.  We  write  the  1  ten  beneath  in  tens' 
place,  and  reserve  the  5  hundreds. 


MULTIPLICATION.  31 

7  times  2  hundreds  are  14  hundreds,  and  14  hundreds  plus  5  hun- 
dreds reserved  are  19  hundreds,  or  1  thousand  and  9  hundreds.  We 
write  the  9  hundreds  beneath  in  hundreds'  place,  and  the  1  thousand 
in  thousands'  place. 

The  product  is  1  thousand  9  hundreds  1  ten  8  ones,  or  1918. 

38.  39.  40.  41. 

Multiplicand     756            4567            1109              6201 
Multiplier         _J  3  _5_  8 

42.  43.  44.  45. 

Multiplicand     3416          2608            12345              24301 
Multiplier         6  ^_2  7  9 

46.  What  is  the  product  of  64.25  by  5  ? 

^^  -  .  ..        T        nA  r,r  \ SoluHon,  —  5  tiuies  5  huu- 

Multipiicand       d4.Jo  t     .^.  or   i.     j     i^i, 

^  dredths  are   25   hundredths, 

^^l^iP^^^^  . ^  or  two   tenths   and   5  hun- 

Product  321.25  dredths.      We  write   the   5 

hundredths  beneath  in  hun- 
dredths' place  and  reserve  the  2  tenths. 

5  times  2  tenths  are  10  tenths,  and  10  tenths  plus  the  2  tenths  re- 
served are  12  tenths,  or  1  one  and  2  tenths.  We  write  the  2  tenths 
beneath  in  tenths'  place,  and  reserve  the  1  one. 

5  times  4  ones  are  20  ones,  and  20  ones  plus  the  1  one  reserved  are 
21  ones,  or  2  tens  and  1  one.  We  write  the  1  one  beneath  in  ones' 
place,  and  reserve  the  2  tens ;  and  so  on.  The  multiplier  being  an 
integer, the  product  has  as  many  decimal  places  as  the  multiplicand. 


Multiply 
By 

47. 

35.07 
8 

48. 

6.135 
5 

49. 

1635.9 

2 

50. 

7128.53 

7 

Multiply 

By 

51. 

$  124.35 
4 

52. 

$  192.547 
6 

53. 

$  823.50 
8 

54. 

$8537.64 
9 

32  MULTIPLICATION. 

ORAL    EXERCISES. 

55.  How  much  will  12  coats  cost  at  $  9  each  ? 

56.  When  plows  are  $  11  each,  what  will  eight  cost  ? 

57.  If  7  men  can  do  a  piece  of  work  in  9  days,  in  what 
time  will  1  man  do  the  same  work  ? 

58.  If  a  vessel  sails  at  the  rate  of  12  miles  an  hour,  how  fai* 
will  it  sail  in  12  hours  ? 

59.  What  will  11  barrels  of  flour  cost  at  $  10  each  ? 

60.  How  many  horses  will  consume  in  one  day  as  many 
bushels  of  oats  as  11  horses  consume  in  12  days  ? 

What  is  the  value  of 

61.  4  X  3  X    6  ?  67.  12  X  10  +    9  ? 

62.  8  X  7  +    9  ?,  68.  11  X  11  -  12  ? 

63.  6  X  4  -  10  ?  69.  11  X    6  -  10  ? 

64.  9  X  5  -    8  ?  70.  11  X  10  -  12  ? 

65.  6  X  5  +  12  ?  71.  10  X    7  -  11  ? 

66.  3  X  4  X  10  ?  72.  12  X    8  +  10  ? 

73.  Give  the  product  of  the  following  numbers  multiplied 
by  12  ;  by  11 ;  by  5  ;  by  4. 

9,  11,  3,  5,  12,  8,  2,  7,  10,  6,  4. 

74.  Multiply  the  numbers  of  the  preceding  exercise  by  8  ; 
by  9  ;  by  7 ;  by  6,  and  add  3  to  each  product. 

75.  How  many  are  6  times  24  ? 

Solution.  —  24  is  20  +  4  ;  6  times  20  are  120;  6  times  4  are  24; 
hence  6  times  20  +  4,  or  24,  are  120  +  24,  or  144. 

76.  At  26  cents  a  yard,  how  many  cents  will  7  yards  of 
cloth  cost  ? 

77.  At  the  rate  of  23  miles  an  hour,  how  many  miles  will  a 
train  of  cars  move  in  8  hours  ? 

78.  What  will  9  suits  of  clothes  cost  at  $  30  a  suit  ? 

79.  What  will  5  sewing-machines  cost  at  $  50  each  ? 


MULTIPLICATION. 


33 


WRITTEN     EXERCISES. 


80.   What  is  the  product  of  534  and  242  ? 

Solution.  —  We 
write  the  factors  so 


534 
242 


Multiplicand  .  .  .       242 
,  Multiplier 534 


1068 
2136 
1068   . 

129228 


Partial 
Products 

Product     . 


129228 


that  the  right-hand 
figure  of  each  shall 
stand  in  the  same 
column. 

Multiplying  by  the 
2  ones,  we  have  1068 
ones  as  the  first  par-  ^ 
tial  product;  multiplying  by  the  4  tens,  we  have  2136  tens  for  the 
second  partial  product,  which  we  write  so  that  its  right-hand  figure 
shall  come  in  the  tens*  column  ;  multiplying  by  the  2  hundreds,  we 
have  1068  hundreds  for  the  third  partial  product,  which  we  write  so 
that  units  of  the  same  order  shall  come  in  the  same  column  ;  adding 
the  three  partial  products,  we  have  129228  as  the  product  required. 

To  prove  the  work,  since  the  product  is  the  same  whatever  the  or- 
der of  the  factors,  we  multiply  the  242by  the  534,  and  have,  as  before, 
129228. 


81. 

82. 

83. 

84. 

Multiply 

763 

1345 

i06 

1621 

By 

37 

45 

26 

34 

85. 

86. 

87. 

88. 

Multiply 

134.7 

17.58 

3.049 

$  25.75 

By 

86 

285 

329 

703 

Note.  —  In  solution  of  Ex.  88,  there  being  Otens  in  the  multiplier,  we  pass  to 
the  hundreds  of  the  nmltiplier. 


59.     Rule  for  Multiplication. 

Write  the  multiplier  under  the  multiplicand,  with  a  line 
beneath. 

Begin7ilnfj  at  the  right,  multiply  each  figure  of  the  multipli- 
cand by  each  figure  of  the  multiplier  successively,  placing  the 


34  MULTIPLICATION. 


right-hand  figure  of  each  partial  product  under  the  figure  of 
the  multiplier  that  produced  it. 

Add  the  partial  products,  and  from  the  right  of  the  result 
'point  off  as  many  decimal  places  as  are  found  in  both  factors. 

60.    Proof.     See   if  the   same  result   is   obtained  by 
multiplying  the  multiplier  by  the  multiplicand. 


Multiply : 

89. 

347  by  769. 

104. 

$  817.42  by  358. 

90. 

826  by  243. 

105. 

8.439  by  125. 

91. 

90.4  by  85. 

106. 

86491  by  688. 

92. 

$  32.13  by  91. 

107. 

49382  by  294. 

93. 

456.7  by  (SS. 

108. 

$  887.95  by  761. 

94. 

8.901  by  542. 

109. 

4963  by  845. 

95. 

$  23.45  by  397. 

110. 

$  28.59  by  927. 

96. 

6789.0  by  645. 

111. 

938.42  by  347. 

97. 

$  198.06*by  805. 

112. 

61904  by  869. 

98. 

45.32  by  907. 

113. 

$329.87  by  35. 

99. 

982.4  by  3004. 

114. 

42935  by  942. 

100. 

$  60.51  by  768. 

115. 

$  864.23  by  346. 

101. 

87.35  by  94. 

116. 

84917  by  809. 

102. 

$  80.42  by  832. 

117. 

173.24  by  935. 

103. 

30.69  by  907. 

118. 

$  98.983  by  871. 

119.  On  board  of  a  steamer  there  are  163  barrels  of  sugar, 
each  weighing  295  pounds.  What  is  the  weight  of  the 
whole  ? 

120.  The  factors  of  a  product  are  1468  and  87  ;  what  is  the 
product  ? 

121.  The  factors  of  a  product  are  681,  507,  and  12  ;  what 
is  the  product  ? 

.122.  The  multiplicand  is  804.51,  and  the  multiplier  63; 
what  is  the  product  ? 

123.  What  number  =  914  08  X  64  ? 

124.  At  $  82.50  an  acre,  what  is  the  value  of  25  acres  oi 
laud  ? 


MULTIPLICATION.  36 

125.    What    is   the  product  of  five  thousand  four   hundred 
fourteen,  and  fifteen  thousandths,  multiplied  by  38  ? 

126.  8304.5    X    77  =  what? 

127.  7038.61  X  126  =  what  ? 

128.  824.84  X  424  =  what  ? 

129.  $  62.005  X    91  =  what  ? 

130.  $  47.168  X  208  =  what  ? 

131.  $  617.43  X  355  =  what  ? 

•   132.   How  many  bushels  of  wheat  can  be  raised  on  5634 
acres  at  the  rate  of  47  bushels  per  acre  ? 

133.  Multiply  nine  hundred  sixty-five,  and  thirteen  hun- 
dredths, by  three  thousand  seven  hundred  five  ? 

The  multiplier  a  number  of  tens,  hundreds,  etc. 

134.  Multiply  48  by  10  ;  by  100. 

4g  43  Solution.  —  10  forty-eights  is  the 

j^Q  ii^QQ  same  as  48  tens  (Art.  58),  or  480. 

—— :  Also,  100  forty-eights  is  the  same 

480  4800  ^g  ^g  himdreds,  or  4800. 

135.  Multiply  427  by  10,  by  100,  and  by  1000,  and  add  the 
results. 

136.  How  many  are  a  thousand  times  7854  ?  138  ?     "What 
is  the  product  of  $  55.56  multiplied  by  100  ? 

^  Pjw  ^r*  Solution.  —  The  removal  of  a  figure  one 

place  to  the  left  in  a  number  increases  the 

value  expressed  ten-fold  (Art.  18).     Hence, 

$  5556.00  ^e  multiply  $  55.56  by  10  by  removing  the 

decimal  point  one  place  to  the  right,  and  by 
10  X  10,  or  by  100,  by  removing  the  point  two  places  to  the  right. 

137.  Multiply  97  by  600. 

Solution.  — 600  is  100  times  6  ;  600  times 
97.  is  the  same  as  100  times  6  times  97.     6 
__—  times  97  are  582,  and  100  times  6  times  97 

58200  are  100  times  582,  or  58200.     That  is,  — 


36  MULTIPLICATION. 

61.    To  multiply  by  a  number  of  tens,  hundreds,  etc. 

Midtiply  without  regard  to  the  ciphers  at  the  right  of  the 
multiplier,  annex  that  number  of  ciph&rs  to  the  product,  and 
give  it  as  many  decimal  figures  as  the  multiplicand  has. 

Multiply : 

138.   814  by  1000.  143.  3450.9  by  800. 

13a    3921  by  70.  144.  $  1604.05  by  2000. 

140.  54.78  by  9000.  145.  446.008  by  4600. 

141.  1342  by  450.  146.  33300  by  820. 

142.  $  611.31  by  110.  147.  $77880  by  300. 

148.  If  the  earth  moves  about  the  sun  at  tbe  rate  of  68000 
miles  an  hour,  how  far  does  it  move  in  240  hours  ? 

149.  A  square  mile  is  640  acres.  How  many  acres  has  a 
State  whose  area  is  7800  square  miles  ? 

MISCELLANEOUS     EXERCISES. 

150.  Bought  20  barrels  of  flour  at  $  8.50  a  barrel,  8  tons  of 
fine  feed  at  $19.50  a  ton,  and  oats  for  $63.25,  and  gave  in 
payment  a  $  500  bank-bill.     How  much  should  be  paid  back  ? 

151.  In  a  certain  battle  an  army  lost  416  killed  and  3  times 
IS  many  wounded.  The  enemy's  loss  was  5  times  as  many. 
What  was  the  entire  loss  of  killed  and  wounded  in  the 
battle  ? 

152.  A  clerk  had  a  salary  of  $  1500  for  a  year  of  52  weeks. 
His  board  cost  him  $4.25  a  week  ;  he  wasted  for  cigars  and 
liquor  $  1.30  a  week  ;  and  his  other  expenses  were  for  the  year 
%  150.     How  much  did  he  save  ?     How  much  could  he  have 

\j  saved  if  he  had  avoided  the  cigars  and  liquor  ? 

153.  Two  trains  of  cars  leave  Boston  at  the  same  time  on 
the  same  railroad  for  the  West.  One  goes  at  the  rate  of  31.50 
miles  an  hour,  and  the  other  at  the  rate  of  16.25  miles  an 
hour.  How  far  apart  will  the  two  trains  be  at  the  end  of  48 
hours  ? 


M 


MULTIPLICATION.  37 


154.    A  drover  has  bought  120  fat  oxen,  each  weighing  on 
an  average  1360  j)ounds.     What  is  the  weight  of  the  whole  ? 
^  I      155.    In  a  school  of  56  pupils  there  are  29  boys  whose  aver- 
age  weight  is  85  pounds.     The  girls  average  79  pounds  each. 
What  is  the  weight  of  the  school  ? 

156.  A  merchant  bought  130  yards  of  cloth  at  $  3.75  a  yard^ 
and  sold  the  whole  for  $  573.95.     How  much  did  he  make  ? 

157.  Multiply  the  sum  of  842  and  796  by  twice  their  differ- 
ence. 

158.  When  boards  are  $31.50  a  thousand  feet,  and  shingles 
$  4.25  a  tlxousand,  what  will  40  thousand  feet  of  boards  and 
22  thousand  of  shingles  cost  ? 

159.  James  Hudson  has  a  house  worth  $  2500,  another 
worth  $  1900,  and-  60  acres  of  land  worth  $75  an  acre.  How 
much  more  is  the  land  worth  than  the  two  houses  ? 

160.  Bought  17  yards  of  silk  at  $  2.75  a  yard,  114  yards  of 
carpeting  at  $  1.80  a  yard,  and  $  17  worth  of  lining.  What 
was  the  amount  of  the  bill  ? 

161.  A  drover  bought  60  oxen  at  $  50  a  head,  120  sheep  at 
$  4.25  a  head,  and  28  cows  at  $  45.50  a  head.  He  returned 
20  of  the  oxen  at  cost  and  bought  5  horses  at  $125  each. 
What  was  the  cost  of  the  whole  to  him  ? 

1/162.  Bought  24  car-loads  of  wheat,  each  car  holding  325 
bushels,  at  $  1.48  a  bushel.  I  sold  the  wheat  at  $  1.65  a 
bushel.     How  much  did  I  gain  ? 

QUESTIONS. 

50.  What  is  multiplication  1  51.  What  is  the  multiplicand?  52. 
The  multiplier  ?     53.    The  product  ?     The  factors  of  the  product  ? 

54.  What  is  the  sign  of  multiplication  ?  55.  What  is  a  concrete 
number  ?     56.   An  abstract  number  ?         " 

58,  What  are  principles  of  multiplication  ?  59.  How  are  num- 
bers written  for  multiplying  1  How  do  you  multiply  ?  60.  What 
is  the  proof  1 

61.    How  do  you  nuiltiply  by  a  number  of  tens,  hundreds,  etc  1 


38  DIVISION. 


DIVISION. 

62.  1.  James  has  32  books.  How  many  times  8  books 
has  he  ? 

2.  How  many  quarts  of  nuts,  at  9  cents  a  quart,  can  be 
bought  for  54  cents  ? 

3.  How  many  times  9  cents  are  54  cents  ?  How  many 
times  9  cents  in  54  cents  ? 

4.  If  7  men  share  42  pounds  of  tea,  how  many  pounds  will 
each  man  have  ? 

5.  Ella  has  27  cents.  How  many  pencils  at  5  cents  each 
can  she  buy,  and  how  many  cents  left  ? 

63.  Division  is  finding  how  many  times  one  number  is 
contained  in  another;  or,  finding  one  of  the  equal  parts  of 
a  number. 

64.  The  Dividend  is  the  number  divided. 

65.  The  Divisor  is  the  number  by  which  we  divide. 

66.  The  Quotient  is  the  result  of  a  division. 

67.  The  Remainder  is  the  part  of  the  dividend  left,  when 
the  latter  does  not  contain  the  divisor  an  exact  number  of 
times. 

68.  The  Sign  of  Division,  -j-  ,  or  :  ,  means  divided  by. 
Thus, 

16  -T-  8,  or  16  :  8,  is  read,  sixteen  divided  by  eight. 
Division  is  also  indicated  by  writing  the  divisor  at  the 
left  of  the  dividend,  with  a  curve,  ),  between,  or  by  writing 
the  divisor  under  the  dividend,  with  a  horizontal  line  be- 
tween.    Thus, 

2)  6,  or  |,  may  be  read,  six  divided  by  two. 


6d.  A  Parenthesis,  (  ) ,  or  Vinculum, ,  is  used  to  in- 
clude such  numbers  as  are  to  be  considered  together. 
Thus,  6  +  8  ^  2  +  5  =  15,  but  (6  +  8)  -f-  (2  +  5)  =  2. 

Note.  —The  signs  X  and  -j-  have  no  force  in  either  direction  beyond  a 
f  or  a  —  unless  a  parenthesis  is  used.  Operations  indicated  by  them  must 
be  performed  first. 

70.  The  Equal  Parts  into  which  a  number  may  be  di- 
vided are  named,  according  to  their  size.     Thus,  — 

One  of  two  equal  parts,  written  |-,  is  called  one  half ; 
One  of  three  equal  parts,  written  ^,  is  called  one  third  ; 
One  of  four  equal  parts,  written  \,  is  called  one  fourth; 
Two  of  three  equal  parts,  written  f,  is  called  two  thirds  ; 
Three  of  four  equal  parts,  written  |,  is  called  three  fourths  ; 
and  so  on. 

ORAL    EXERCISES. 

6.  John  has  84  cents.     How  many  times  7  cents  has  he  ? 

7.  How  many  times  12  cents  in  84  cents  ? 

8.  Ellen  has  49  peaches.  Should  she  divide  them  equally 
among  7  of  her  playmates,  how  many  would  each  receive  ? 

9.  What  number  is  one  of  the  7  equal  parts  of  49  ?  What 
is  -V-  ? 

10.  If  $  96  he  equally  divided  among  12  men,  what  sum 
would  each  receive  ?     How  much  is  ^^  of  }>  96  ? 

11.  In  84  days  how  many  weeks  ?  How  many  times  7  in 
84  ?     What  is  4^  of  84  ? 


What  is 

12. 

13. 

14. 

15. 

16. 

63-7? 

8)  72? 

72:9? 

w 

I  of  36  ? 

64-^  8? 

9)  81? 

96  :  12  ? 

1.0  8    ? 

i  of  45  ? 

54-^9? 

5)  60? 

99  :  11  ? 

A^-? 

1  of  96  ? 

49  -^  7  ?        11)  66  ?         32  :  4  ?         -V"  ^         tV  oi  110  ? 
24  ~  8  ?       10)  80  ?         63  :  9  ?         i,^  ?       ^  oi  132  ? 


40  DIVISION. 

17.  Give  the  quotients  of  the  numbers  in  the  following  line 
divided  by  2  ;  by  4  ;  by  6  ;  by  8  ;  by  10. 

8,     12,     16,     20,     24,     32,     36,     48,     60,     64. 

18.  Give  the  quotients  of  the  numbers  in  the  following  line 
divided  by  3  ;  by  5 ;  by  7 ;  by  9  ;  by  8. 

28,     33,     35,     42,     49,  63,     81,     96,     100. 
Supply  the  necessary  numbers  in  the  following : 

19.  20. 

6  X    ?  ==  42  11  X  ?  ==  121 

9  X    ?  ==  108  ?  X  8  =  64 

?  X  12  =  72  7  X  ?  :=  42 

4x?=^32  ?x9r=72 

21.  22. 

3  X  4  X  ?  =  84  -8/-  +  9  =  ? 

2  X  ?  X  6  =  144  132  :  ?  =  12 

3  X  3  X  ?  =  81  144  :  ?  =  12 
3  X  3  X  ?  =  27  ^-fii  -{-5  =  ? 

2*3.    In  24  how  many  times  2?6?8?4?12? 

24.  In  45  how  many  times  5?3?9?15? 

25.  In  60  how  many  times  5  ?  10  ?  6  ?  4  ?  12  ?  15  ?  20  ? 

26.  At  8  cents  each,  how  many  cocoa-nuts  can  be  bought 
for  88  cents  ? 

27.  If  a  dozen  of  eggs  cost  48  cents,  what  does  one  cost  ? 

28.  How  many  pounds  of  meat  can  be  bought  for  72  cents 
at  8  cents  a  pound  ? 

29.  How  many  yards  of  cloth  at  $  6  a  yard  will  pay  for  8 
tons  of  coal  at  $  9  a  ton  ? 

30.  If  4  men  do  a  piece  of  work  in  12  days,  how  long  will  it 
take  6  men  ? 

31.  What  number  divided  by  9  will  give  6  ? 

32.  If  4  cords  of  wood  cost  $  36,  what  will  7  cords  cost  ? 

33.  How  many  twelve-quart  cans  will  hold  84  gallons  of 
milk? 

34.  If  a  train  of  cars  runs  108  miles  in  9  hours,  what  is  the 
rate  per  hour  ? 


Divisioisr,  ^1 

71.  Principles  of  Division. 

1.  The  dividend  is  the  product  of  the  divisor  and  the 
quotient 

2.  When  the  divisor  and  the  dividend  are  like  numbers,  the 
quotient  will  be  an  abstract  number. 

3.  When  the  divisor  is  an  abstract  7iumber,  the  dividend 
and  the  quotient  will  be  like  numbers. 

WRITTEN    EXERCISES. 

35.   Divide  8574  by  6. 

Divisor   6)  8574     Dividend.  Solution.— Wq  write  the 

1429     Quotient.  '^|^|^^^  ^^  *^^  ^^^'^  ^^'  ^^^ 

P  dividend,  with  a  curve  be- 

tween  them,   and  begin  at 

Proof  8574  the  left  to  divide. 

6  in  8  thousands,  1  thou- 
sand times,  with  2  thousands,  or  20  hundreds,  remaining.  We  write 
the  1  thousand  beneath  in  thousands'  place. 

Uniting  the  20  hundreds  with  the  5  hundreds  of  the  dividend,  we 
have  25  hundreds.  6  in  25  hundreds,  4  hundred  times,  with  1  hun- 
dred, or  10  tens,  remaining.  We  write  the  4  hundreds  beneath  in 
hundreds'  place. 

Uniting  the  10  tens  with  the  7  tens  of  the  dividend,  we  have  17 
tens.  6  in  17  tens,  2  tens  times,  with  5  tens,  or  50  ones,  remaining. 
We  write  the  2  tens  beneath  in  tens'  place. 

Uniting  the  50  ones  with  the  4  ones  of  the  dividend,  we  have  54 
ones.  6  in  54  ones,  9  times.  We  write  the  9  ones  beneath  in  ones' 
place. 

The  quotient  is  1  thousand  4  hundreds  2  tens  9  ones,  or  1429. 

To  prove  the  work,  we  multiply  the  quotient  by  the  divisor,  and 
have  as  the  product  the  dividend  (Art.  71). 

36.  37.  38.  39. 

4)  1532    7)  5747    5)  4855    8)  9136 

40.         41.         42.  43. 

6)  351.6    8)  97.84    7)  83.30    9)  40.203 


42  DIVISION. 


44. 


f..  r^orro  Solution.  —  Dividing,  there  is  a  final  re- 

^  mainder  of  3.     The  division  of  the  3  we 

Quotient     1274§  indicate  by  -|,  and  writing  this  as  a  part 

5  of  the  result,  we  have  the  quotient  1274|. 

Q'^jO  To  prove  the  work,  we  multiply  the  in- 

o  teger  of  the  quotient  by  the  divisor,  and 

adding  the  remainder,  have  the  dividend. 

Proof  6373 

45.  46.  47.  48. 

5)  3576    12)  8143     8)  4899     7)  58965 

49.  50.  51.  52. 

4)  2693  3)  40.02  6)  364.27  9)  39.137 

53.  How  many  tons  of  coal  at   $  5  a  ton  can  be  bought  for 
$  2025  ? 

54.  How  far  must  a  vessel  sail  each  day  to  sail  1438  miles 
in  7  days  ? 

55.  Paid  for  8  yards  of  broadcloth  $  43.60.     What  was  the 
cloth  a  yard  ? 

56.  If  the  dividend  is  20818,  and  the  divisor  9,  what  is  the 
quotient  ? 

ORAL   EXERCISES. 

57.  How  many  times  is  ^  of  32  contained  in  J  of  60  ? 

58.  The  product  of  two  numbers  is  56,  and  one  of  the  num- 
bers 8.     What  is  the  other  number  ? 

59.  If  the  dividend  is   72,  and  the  divisor  9,  what  is  the 
quotient  ? 

60.  Mr.  Jones  can  earn  in  a  month  $  81,  and  his  son  one 
ninth  as  much.     How  much  can  his  son  earn  ? 

61.  A  farm  of  120  acres  has  been  divided  into  12  lots.    How 
many  acres  in  each  lot  ? 

62.  What  is  i  of  64  ?     i  of  81  ?     -,V  of  JIO  ?     ^ij  of  96  ? 

63.  One  man  can  do  a  piece  of  work  in  132  days.     In  what 
time  can  11  men  do  it? 


'    DIVISION.  43 

64.  A  man  earned  $  144  in  12  weeks.  What  was  that  a 
week  ? 

65.  How  many  lO's  in  100  ?     How  many  lOO's  in  1000  ? 

66.  In  65  how  many  tens  and  what  over  ?  In  235  how 
many  hundreds  and  what  over  ? 

67.  How  many  times  20  in  100  ?  How  many  timen 
(20  -^  4)  in  (100  -^  4)  ?  How  many  times  (20  X  2)  in 
(100  X  2)  ? 

72.    A  General  Principle  of  Division. 

Dividing  or  Tnultiplying  both  dividend  and  divisor  by  the 
same  number  does  not  alter  the  quotient. 

WRITTEN    EXERCISES. 

68.  Divide  1185  by  12. 

98^9^  Quotient.  Solution.  — \2  is  in  118 

Divisor   12  )1185       Dividend.         tens,  9    tens    times.      We 
"I^Qg  .  write  the  9  tens  over  the 

"TT^p.  tens  of  the  dividend. 

9  tens  times  12  are  108 

__  vtens,  which  taken  from  the 

9  118    tens  of  the   dividend 

leaves  10  tens,  and  uniting 

with  these  10  tens  the  5  ones  of  the  dividend,  we  have  105  ones.     12 

is   in  105,  8  times.     We  write  the  8  ones  above  the  ones  of  the. 

dividend. 

8  times  12  are  96,  which  taken  from  the  105  of  the  dividend  leaves 
9,  and  indicating  its  division  by  the  divisor,  we  have  ^\.  Writing 
the  y^Tj  as  a  part  of  the  quotient,  we  have  as  the  quotient  98^^^. 

69.  Divide    9876  by  21.  72.   Divide  106.97  by  19. 

70.  Divide    3276  by  14.  73.   Divide  26.575  by  25. 

71.  Divide  31278  by  23.  74.   Divide  1868.5  by  37. 

73.    Rule  for  Division. 

Write  the  divisor  at  the  left  of  the  dividend  with  a  curve 
between  them. 

Find  how  many  tiiues  the  divisor  is  contained  in  the  few* 


44 


DIVISION, 


est  left-hand  figures  of  the  dividend  that  will  contain  it,  and 
write  the  result  under  or  over  the  right-hand  figure  of  the  div- 
idend used,  as  the  first  quotient  figure. 

Multiply  the  divisor  by  this  quotient,  subtract  the  product 
from  the  partial  dividend  u^ed,  and  to  the  rew.ainder  annex 
the  next  dividend  figure  for  a  second  partial  dividend. 

Divide  and  proceed  as  before,  until  all  the  dividend  figures 
lave  been  used. 

If  there  is  a  final  remainder,  write  it  with  the  divisor 
under  it,  05  a  part  of  the  quotient. 

Note  1.  —  The  division  is  called  Short  Division  when  only  tlie  divisor,  div- 
idend, and  quotient  are  written;  and  Long  Division  when  each  process  of  the 
solution  is  written. 

In  short  division  the  quotient  is  usually  written  below  the  dividend,  and  in 
long  division  over,  or  at  the  right  of,  the  dividend. 

Note  2.  —  When  the  divisor  is  an  integer,  and  the  quotient  is  written  under 
or  over  the  dividend,  the  decimal  point  in  the  quotient  must  be  placed  immedi- 
ately under  or  over  the  point  in  the  dividend. 

74.  Proof.  —  Multiply  the  quotient  by  the  divisor,  and 
to  the  product  add  the  remainder,  if  any.  The  result 
should  be  the  dividend. 


75.   Divide  182.72  by  45,  and  prove  the  work. 


Solution. 

Proof. 

4.06A 

Quotient. 

4.06^  Quotient. 

Divisor    45)182.72 

Dividend. 

45      Divisor. 

180 

2030 

2.72 

1624 

2.70 

182.70 

.02 

Remainder. 

.02     Remainder. 
182.72     Dividend. 

76.  Find  the  quotient  of  $  114.48  divided  by  36. 

77.  Find  one  twenty-eiglitli  of  $  890.56. 


DIVISION.  45 

73.  Divide  347692  by  351.  85.  1298763  H- 873  :=  ? 

79.  Divide  8468.31  by  793.  86.  -^Ij  of  63450  =  ? 

80.  Divide  947684  by  982.  87.  4127.098  -^-  2007  =  ? 

81.  Divide  3287.64  by  735.  88.  8643.21  -f-  987  =  ? 

82    Divide  6798.341  by  1234.  89.   (643  x  857)  -^  456  =  ? 

83.  Divide  $8392.476  by  987.  90.    (984  X  895)  -^  359  :^  ? 

84.  Divide  3004760  by  5942.  91.   ^^^J/-^^  =  ? 

92.  Wbat  is  tbe  price  of  one  sleigh  if  85  cost  $10625  ? 

93.  How  muny  horses  at  $225  each  will  $5400  buy  ? 

94.  If  $671178.90  is  the  valuation  of  a  hamlet  having  98 
inhabitants,  what  is  the  average  valuation  to  an  inhabitant  ? 

95.  If  the  distance  across  the  Atlantic  Ocean  is  3000  miles, 
in  how  many  days  will  a  ship,  sailing  115  miles  a  day,  make 
that  distance  ? 

96.  What  number  multiplied  by  512  gives  a  product  of 
1763.68  ? 

97.  What  number  multiplied  by  2135  gives  a  product  of 
6419945  ? 

98.  Divide  49300  by  432.  100.    $5711.04-^108==? 

99.  Divide  700074  by  2047.      101.    $  50000  -^  365  =  ? 
102.    Divide  489  by  25  to  hundredths. 

19.56 


25)  489.00 
25_ 

239 
225 


Solution.  — A^^  =z  489.00.  Dividing  the 
489.00  by  25,  we  obtain  as  theresidt  required 
19.56. 

If  the  division  had  not  terminated,  we  could 

14,0  have  denoted  this  by  placing  the  sign  -\-  after 

12.5  the  hundredths  in  the  quotient  to  indicate  the 

7~^  incompleteness. 


1.50 

103.  Divide  2722.5  by  44  to  hundredths. 

104.  Divide  9164  by  86  to  thousandths. 

105.  Divide  $  12625  by  404  to  cents. 

106.  Divide  17552  by  128  to  thousandths. 


n!> 


46  DIVISION. 

107.  A  gentleman  dying  left  an  estate  valued  at  on<^  mit 
lion  dollars  to  be  equally  divided  among  thirty-three  heirs. 
What  did  each  receive  ? 

The  divisor  a  number  of  tens,  hundreds,  etc. 
lOa    Divide  1467  by  10  ;  by  100  j  by  1000. 

1467.  -^      10  =  146.7  Solutio7i,  —  The  removal  of  a 

1467.  -^    100  =  14.67  figure  one  place  to  the  right  de- 

1467.  -L.  1000  ==  1.467  creases  its  vakie  ten  times  (Art. 

18).  To  divide  a  number  by  10, 
or  to  find  1  tenth  of  a  number,  it  is  only  necessary  to  move  each 
figure  of  it  one  place  to  the  right.  This  is  done  by  moving  the  deci- 
mal point  one  place  to  the  left,  giving  14:6^^  as  the  quotient.  In  the 
same  way  two  removals  of  the  point  to  the  left  divides  by  100,  three 
removals  by  1000,  and  so  on.     That  is, 

75.    To  divide  by  10,  100,  1000,  etc., 

Move  the  decimal  point  of  the  dividend  as  many  2^l(^ces  to 
the  left  as  there  are  ciphers  in  the  divisor. 

109.   Divide  5824  by  160. 

Solution.  — \Wh\Q  X 
10.  Dividing  both  divi- 
sor and  dividend  by  10, 
by  removing  the  decimal 
point  one  place  to  the 
left,  the  division  becomes 
the  same  as  582.4  by  16, 
which  gives  as  the  quo- 
tient  .36.4. 
Or,  if  the  exact  remainder  is  required,  we  may  indicate  the  division 
by  10,  by  marking  off  one  place,  and  have  as  a  quotient  582,  and  a  re- 
mainder of  4  ones.  Dividing,'  582  by  16,  we  have  for  a  quotient  36, 
and  a  remainder  6  tens.  Uniting  the  two  remainders,  we  have  as  the 
true  remainder  64. 


36.4 

16.0)  682.4 
48 

36|tVj 
1610)  582|4 
48 

102 

or            102 

96 
6.4 

96 
64 

6.4 

V 


DIVISION.  47 

110.  There  are  1000  mills  in  a  dollar ;  how  many  dollars 
are  there  in  19650  mills  ? 

111.  Divide  3962  by  100  ;  by  1000. 

112.  Divide  9108  by  200  to  hundredths. 

113.  Divide  12386  by  90,  and  find  the  true  remainder. 

114.  Divide  5886990  by  5400  to  hundredths. 

115.  In  the  State  of  Alabama  143727  acres  have  pro- 
duced 51000  bales  of  cotton.     What  was  that  for  a  bale  ? 

116.  If  light  moves  at  the  rate  of  186000  miles  in  a  second, 
how  long  is  it  in  passing  from  the  sun  to  the  earth,  a  distance 
of  92000000  miles? 

Divide: 

117.  8496453  by  8291.  125.   849  x  863  by  1252. 

118.  2907654  by  3782.  126.   84  x  96  X  25  by  1189. 
119.' 72659302  by  1234.        127.   694^87  +  956  by  609. 

120.  67890123  by  5678.  128.  847  X  12  X  900  by  9. 

121.  4567.890  by  2961.  129.  9008  x  7080  by  2090. 

122.  184.837  by  349.  130.  $  945.65  by  850. 

123.  6239076  by  6384.  131.  843  -  159  by  29  X  7. 

124.  1689.783  by  945.  132.  $  849.625  by  975. 

MISCELLANEOUS    EXERCISES. 

133.  A  farmer  obtained  $7665  for  the  apples  from  his 
orchard  at  $  3  a  barrel.     How  many  barrels  did  he  have  ? 

134.  Two  men  start  from  the  same  place  and  travel  in 
opposite  directions,  one  at  the  rate  of  25  miles  a  day,  and  the 
other  at  the  rate  of  31  miles  a  day.  When  5656  miles  apart 
how  many  days  had  they  traveled  ? 

135.  A  cargo  of  25  tons  of  coal  was  bought  by  the  long  ton 
of  2240  pounds,  and  sold  by  the  short  ton  of  2000  pounds. 
How  many  short  tons  was  the  gain  ? 

136.  A  farmer  bought  75  acres  of  land  at  $  34  an  acre,  and 
85  acres  at  $  20.40  an  acre.  What  was  the  average  cost  of 
the  whole  an  acre  ? 


48  DIVISION. 

\J  137.  If  a  man  can  save  $  15  in  each  of  the  12  months  of  a 
year,  in  how  many  years  can  he  save  enough  to  amount  to 
12520? 

138.  The  product  of  two  factors  is  96000,  and  one  of  the 
factors  is  150.     What  is  the  other  factor  ? 

139.  Bought  a  quantity  of  boards  for  $  2650,  and  sold  the 
same  for  $  3286,  and  thereby  gained  $  6  a  thousand  feet.  How 
many  thousand  feet  were  there  ? 

140.  The  product  of  three  factors  is  33600.  Two  of  the 
factors  are  15  and  35.     What  is  the  third  ? 

141.  Texas  contains  274400  square  miles  and  Massachu- 
setts 7800.  How  many  States  of  the  size  of  Massachusetts 
might  be  made  out  of  Texas,  and  how  many  square  miles 
over  ? 

142.  If  the  divisor  is  350  and  the  dividend  262500,  what  is 
the  quotient  ? 

143.  A  man  bought  164  acres  of  land  for  $  80  an  acre,  and 
sold  a  part  for  $  4480,  at  the  same  rate.  How  many  acres 
had  he  then  left  ? 

144.  If  the  dividend  is  71142  and  the  quotient  1002,  what 
is  the  divisor  ? 

145.  A  man  exchanged  159  cords  of  wood  at  $5  a  cord, 
for  a  horse  valued  at  $  144,  and  the  balance  in  sheep  at  $  3 
apiece.     How  many  sheep  did  he  receive  ? 

QUESTIONS. 

63.  What  is  division  ?  64.  What  is  the  dividend  ?  65.  The  di- 
visor?    66.   The  quotient?     67.   The  remainder ? 

68.  What  is  the  sign  of  division  ?  69.  What  is  a  parenthesis,  or 
vinculum,  used  to  include  ? 

71.  What  are  principles  of  division  ?  72.  What  is  a  general  prin- 
ciple of  division  ? 

73.  How  are  the  numbers  written  for  dividing?  How  do  you  di- 
vide ?  How  is  a  final  remainder  written  ?  74.  What  is  the  proof  of 
dtvision  ? 


REVIEW.  49 


REVIEW. 

ORAL     EXERCISES. 

76.    1.   What  number  added  to  18  will  make  27  ? 

2.  John  had  37  cents.  He  gave  7  to  his  brother,  and 
shared  the  remainder  equally  with  his  two  sisters.  What 
was  his  share  ? 

3.  A  farmer  had  18  chickens.  The  foxes  killed  3  and  he 
sold  7.     How  many  has  he  left  ? 

4.  Thomas  gave  17  cents  for  a  whip,  13  cents  for  a  slate, 
12  cents  for  a  copy-book,  and  10  cents  for  pencils.  How  much 
change  did  he  receive  from  a  dollar  bill  ? 

5.  (5x6  + 5) -^(14-7)  =  ? 

6.  John  had  30  marbles.  He  lost  half  of  them  and  then 
gave  away  5.     How  many  had  he  left  ? 

7.  A  man  had  $  57  ;  he  lost  at  one  time  $  5,  and  at  another 
$  10.     How  many  had  he  left  ? 

8.  Bought  7  tons  of  coal  at  $  8  per  ton,  and  gave  in  payment 
2  twenty-dollar  bills  and  2  ten-dollar  bills.  How  much  change 
should  be  received  back  ? 

9.  I  have  3  bags  of  nuts.  There  are  2  bushels  in  each  bag. 
What  is  the  whole  worth  at  $  3  a  bushel  ? 

10.  George  had  in  a  basket  41  apples.  He  gave  5  to  Susan, 
7  to  Lucy,  and  6  to  Henry.  What  is  the  value  of  those  left 
at  2  cents  each  ? 

Find  the  result  of : 

11.  31  _  9  +  (6  X  3).  15.  (30  -f  25  -  23)  X  3. 

12.  (63T9  ~  12)  X  (14  -  8).  16.  46  -  16  +  10  --  10. 

13.  (irx  8  -  16)  ~  9.  17.  (81  -^  9)  X  (75  -^  25). 

14.  65  ^  (60  -^  5  4-  30).  is.  100  -  (15  X  5) -^  5. 

19.  Two  men  start  from  the  same  place  and  go  in  the  same 
direction ;  one  travels  at  the  rate  of  5  miles  an  hour,  and  the 


50  REVIEW. 

other  at  the  rate  of  9  miles  an  hour.     How  far  apart  will  they 
be  in  12  hours  ? 

20.  A  steamboat  can  run  10  miles  an  hour  down  stream, 
and  8  miles  up  stream.  After  running  down  stream  4  hours, 
how  long  will  it  be  in  returning  ? 

21.  If  you  should  earn  $  60  in  10  weeks,  and  pay  of  your  earn- 
ings $3  a  week  for  board,  how  much  will  your  net  earn- 
ings be  ? 

22.  William  had  98  peaches.  He  gave  18  to  his  brother, 
and  shared  the  remainder  equally  with  9  others.  How  many 
did  he  give  away  in  all  ? 

23.  If  6  men  can  do  a  piece  of  work  in  8  days,  in  how  many 
days  can  4  men  do  it  ? 

24.  In  what  time  can  6  men  excavate  a  cellar,  which  14 
men  can  excavate  in  3  days  ? 

25.  If  15  men  can  build  a  wall  in  10  days,  in  what  time 
can  25  men  do  it  ? 

26.  If  9  barrels  of  flour  are  worth  $  81,  what  are  7  barrels 
worth  ? 

27.  When  24  pounds  of  coffee  can  be  bought  for  $  6,  how 
many  pounds  can  be  bought  for  $  8  ? 

28.  If  a  man  can  earn  $  96  in  8  weeks,  in  what  time  can  he 
earn  $  60  ? 

29.  A  man  having  $  95  bought  5  coats,  and  had  $  20  left. 
What  did  the  coats  cost  apiece  ? 

30.  Two  men  start  132  miles  apart  and  travel  toward  each 
other,  one  at  the  rate  of  6  miles  an  hour,  and  the  other  at  the 
rate  of  5  miles  an  hour.  How  many  miles  must  each  travel 
before  meeting  ? 

31.  A  cistern  can  be  emptied  in  16  minutes  by  5  pipes.  In 
what  time  can  it  be  emptied  by  only  2  pipes  ? 

32.  A  and  B  start  together  and  travel  in  the  same  direc- 
tion, A  traveling  21  miles  a  day  and  B  27  miles.  When 
they  have  traveled  9  days,  how  much  less  than  60  miles  are 
they  apart  ? 


REVIEW.  51 


WRITTEN   EXERCISES. 

33.  The  area  of  Illinois  is  55405  square  miles,  and  of  Ar- 
kansas 52198  square  miles.  How  many  square  miles  is  Illi- 
nois larger  than  Arkansas  ? 

34.  The  minuend  is  74760,  and  the  subtrahend  34943. 
What  is  the  difference? 

35.  The  difference  between  two  numbers  is  4004,  and  the 
greater  number  is  5496.     What  is  the  smaller  ? 

36.  A  man  had  $  5000.  He  expended  for  a  stable  $  2560.75, 
and  for  a  horse  and  carriage  $  375.87.     What  had  he  left  ? 

37.  The  multiplicand  is  664  and  the  multiplier  19.  What 
is  the  product  ? 

NJ  38.    Six  men  bought  some  property  for  $  5670  and  sold  it 

for  $  7896.84.     What  was  each  man's  share  of  the  gain  ? 
Sj    39.   A  merchant  bought  3  casks  of  sugar,  each  weighing  255 
^pounds,  at  9  cents  a  pound,  and  sold  it  at  11  cents  a  pound. 
How  many  dollars  did  he  make  ? 

40.  A  man  sets  out  to  travel  223  miles  at  the  rate  of  27 
miles  a  day.  When  he  has  gone  61  miles,  in  how  many  days 
can  he  finish  the  distance  ? 

41.  What  is  the  cost  of  365  parlor  organs  at  $  97  each  ? 

42.  The  divisor  is  48,  the  quotient  596,  and  the  remainder 
10.     What  is  the  dividend  ? 

43.  Expended  in  goods  $  A2.bl,  buying  17  yards  of  cloth  at 
15  cents  a  yard,  46  pounds  of  coffee  at  28  cents  a  pound,  16 
gallons  of  molasses  at  76  cents  a  gallon,  and  107  pounds  of 
confectionery.     How  much  was  the  confectionery  a  pound  ? 

I      44.   Bought  310  tons  of  coal  for  $  1472.50  and  sold  it  foi 
\J  1 1549.60.     How  much  did  I  make  a  ton  ? 

45.  Bought  two  lots  of  land  ;  the  first  containing  144  acres 
at  $  12  an  acre,  and  the  second  108  acres  at  $  15  an  acre.  I 
sold  both  lots  at  $  18  an  acre.     What  was  the  gain  per  acre  ? 

46.  The  multiplicand  is  407,  and  the  product  10989.  What 
is  the  multiplier  ? 


52  EEVIEW. 

47.  Bought  288  barrels  of  flour  for  $  1728,  and  sold  it  at  a 
profit  of  $  576.     What  did  I  get  a  barrel  for  it  ? 

48.  A  gentleman  bought  a  house  for  $  5100,  and  farm  stock 
to  the  amount  of  $>  715.80.  He  paid  at  one  time  $  2013,  and 
at  another  $  1981.95.       How  much  remained  to  be  paid  ? 

4     49.      (194  +   65)    X   7  +   Il-^2-f^2  20    _  952  ::::,  ? 

50.  Bought  500  barrels  of  flour  at  15.75  a  barrel,  47  hun 
dred-weight  of  cheese  at  $  9.25  a  hundred,  and  15  barrels  ot 
pork  at  $  21.50  a  barrel.    What  was  the  amount  of  the  whole  ? 

51.  Bought  molasses  for  $  9212,  and  sold  it  at  $  67  a  hogs- 
.head,  and  gained  $  20.     How  man}^  hogsheads  were  there  ? 
L"^^.   A  cargo  of  125  tons  of  coal  was  bought  by  the  long  ton 

of  2240  pounds,  and   sold  by  the   short  ton  of  2000  pounds. 
How  many  long  tons  was  the  gain  ? 

53.  I  have  $  2973.  I  wish  to  invest  it  in  as  many  horses 
as  I  can  at  $  150  each,  and  the  remainder  in  a  carriage.  How 
many  horses  can  I  buy,  and  what  can  I  pay  for  the  carriage  ? 

54.  -7  5f  J-  +  549  -  (2128  4-  7)  X  2  =  what  ? 


y 


55.  (194  +  65)  X  7  +  (352  -  220)  -f- 11  -  952  -^  (91  -  35) 
—  ? 

56.  A  man  bought  360  acres  of  land  at  $  45.50,  and  paid 
down  $  1368.  He  sold  125  acres  at  $  63  an  acre,  and  made 
another  payment.     How  much  did  he  then  owe  for  the  land  ? 

57.  The  product  of  three  numbers  is  40800.  One  of  the 
numbers  is  150,  and  another  16.     What  is  the  third  number  ? 

58.  The  dividend  is  18988  and  the  quotient  2bj\^^.  What 
is  the  divisor  ? 

59.  If  I  have  a  garden  320  feet  long  and  half  as  wide^ 
how  many  times  must  I  walk'  around  it  to  travel  100  miles 
of  5280  feet  each? 

60.  A  man  has  11496,  which  he  wishes  to  lay  out  in  pur- 
chasing cows  and  oxen,  an  equal  number  of  each.  If  he  should 
pay  $  37  for  each  cow  and  $51  for  each  ox,  how  many  of  each 
can  he  buy  ? 


REVIEW.  53 

61.  If  216  pianos  cost  $  112320,  what  will  519  cost  ? 

62.  How  many  pounds  of  coffee,  at  38  cents  a  pound,  will 
pay  for  2  hogsheads  of  sugar  containing  1160  pounds  each. 
at  19  cents  a  pound  ? 

63.  A  farmer  having  $  3038,  bought  15  tons  of  hay  at  $  11 
per  ton,  3  yoke  of  oxen  at  $  155  each,  375  sheep  at  $  5  each, 
and  spent  the  rest  for  cows  at  $  41  apiece.  How  many  cows 
did  he  buy  ? 

64.  A  man  in  his  will  gave  to  each  of  his  2  sons  $  7600 ; 
to  a  third  son  $  1500 ;  to  each  of  3  daughters  $  3775,  and  the 
balance  $6877  to  his  wife.  His  wife  died,  however,  and  the 
whole  property  was  divided  equally  among  his  children  ;  what 
did  each  receive  ? 

65.  The  expenses  of  a  picnic  party  of  9  gentlemen  and  8 
ladies  were  $  2.40  each.  The  gentlemen  paid  all  the  expenses. 
What  did  each  pay  ? 

66.  Sold  160  tons  of  coal  at  $  5  per  ton,  and  a  number  of 
tons  at  $  3  per  ton.  The  value  of  all  the  coal  sold  was  $  965. 
How  many  tons  were  there  ? 

67.  Bought  960  acres  of  land  for  $  12000.  Sold  i  of  it  at 
$  12  per  acre,  J  of  it  at  $  15  per  acre,  and  the  remainder  for 
$  20  per  acre.     Did  I  gain  or  lose,  and  how  much  ? 

REVIEW  QUESTIONS. 

2.  What  is  a  number?  21.  What  is  an  integer?  55.  What  is  a, 
concrete  number  ?    56.  An  abstract  number  ? 

4.  What  is  notation  ?  5.  Numeration  ?  18.  What  are  principles 
of  notation?  31.  What  is  addition?  37.  What  are  principles  of 
addition?  41.  What  is  subtraction?  46.  What  are  principles  of 
subtraction?      , 

50.  What  is  multiplication  ?  58.  What  are  principles  of  multipli- 
cation? 63.  What  is  division  ?  71.  What  are  principles  of  divi- 
sion ? 


54  FACTORS. 


FACTOBS. 

77.  1.  What  two  numbers  multiplied  together  produce  2  ? 
3  ?  5  ? 

2.  What  numbers,  other  than  itself  and  1,  will  divide  6 
without  a  remainder  ? 

3.  What  number  multiplied  by  2  will  produce  6  ? 

4.  Kame  all  the  numbers,  other  than  itself  and  1,  which 
will  divide  21  without  a  remainder. 

5.  What  number  multiplied  by  3  will  produce  21  ? 

6.  Name  the  numbers  between  1  and  15  which  are  the  pro- 
duct of  two  or  more  numbers  greater  than  1. 

78.  The  Factors  of  a  number  are  the  integers  which 
produce  the  number  when  multiplied  together.     Thus, 

3  and  5  are  factors  of  15,  and  2,  3,  and  3  are  factors  of  18. 

79.  A  Composite  Number  is  a  number  having  other  fac- 
tors than  itself  and  one.     Thus, 

4,  6,  8,  9,  and  10  are  composite  numbers. 

Note.  —  One  number  is  said  to  be  divisible  by  another  wlien  there  is  no  re- 
mainder in  the  division.     Thus, 

A  composite  number  is  divisible  by  any  of  its  factors. 

80.  A  Prime  Number  is  a  number  having  no  other  fac- 
tors than  itself  and  one.     Thus, 

1,  2,  3,  5,  7,  11  are  prime  numbers, 

81.  A  Prime  Factor  is  a  factor  that  is  a  prime  number. 

Note.  ~  As  every  integer  is  a  factor  of  itself,  and  has  one  as  the  other  fac- 
tor,  in  speaking  of  the  factors  of  a  number  we  usually  exclude  tlie  number 
itself  and  one. 


FACTORS.  55 

82.    Every  prime  number,  except  2  and  5,  has  1,  3,  7, 

or  9  for  its  unit  figure. 

Note.  —  2  is  a  factor  of  any  number  ending  in  0,  or  whose  ones  are  divisible 
by  2.  3  or  9  is  a  factor  of  any  number  the  sum  of  whose  figures  is  divisible  by 
3  or  9.     5  is  a  factor  of  any  number  ending  in  0  or  6. 

WRITTEN    EXERCISES. 

7.  Find  the  prime  factors  of  56, 

* 

2)  56  Solution,  —  Dividing  by  the  prime 

2)  28  mimbers  2,  2,  and  2,  the  last  quotient 

-—  obtained  is  7,  which  is  also  a  prime 

^ -.  number.     The  factors  required  are 

7  2,  2,  2,  7  ;  or  2^  7. 
Ans.  2,  2,  2,  7  ;  or  2\  7. 

8.  What  are  the  prime  factors  of  84  ?     Of  144  ?     Of  160  ? 

9.  What  are  the  prime  factors  of  462  ?    Of  576  ?    Of  1008  ? 

83.     Rule  for  finding  the  Prime  Factors  of  a  Number. 

Divide  the  given  number  by  any  prime  number  above  one 
that  will  exactly  divide  it  Divide  the  quotient,  if  composite^ 
in  like  manner,  and  so  continue  until  a  prime  quotient  is 
found.  The  several  divisors  and  the  last  quotient  will  be  the 
prime  factors. 

Find  the  prime  factors  of 

10.  210  14.  1155  18.  16028 

11.  2772  15.   2800  19.  17199 

12.  426  16.   3420  20.  10323 

13.  6105  17.   7800  21.  12496 

22.  What  is  the  largest  prime  factor  of  1184  ? 

23.  Find  the  prime  factors  of  4389. 

24.  Find  the  prime  factors  of  6300. 

25.  What  prime  numbers  multiplied  together  produce 
40579  ? 


56  FACTORS. 


CANCELLATION. 

26.  What  is  the  quotient  of  63  divided  by  21  ? 

27.  What  is  one  seventh  of  63  divided  by  ]~  of  21  ? 

28.  What  is  the  quotient  of  J  of  63,  divided  by  J  of  21  ? 

29.  What  is  the  quotient  of  5  X  3  X  2,  divided  by  2  X  3  *? 
Of  3  X  5,  divided  by  3  ?     Of  2  X  5,  divided  by  2  ? 

84.     Cancellation  is  striking  out- the  same  factor  from 
both  dividend  and  divisor. 

85.    Principles  of  Cancellation. 

1.  Striking  out  a  factor  of  a  number  divides  the  number  by 
the  factor. 

2.  Striking  out  the  same  factors  from  dividend  and  divisor 
does  not  affect  the  quotient. 

WRITTEN    EXERCISES. 

30.  Divide  3  times  80  by  48. 

80  X3  ^0X5X^       f.  Solution.  —  Indicating  the 

To        ^^       -*^       q     "  division,   and   canceUng,    by 

striking  out  in  both  dividend 
and  divisor  the  factors  16  and  3,  common  to  both,  there  remains  only 
!ihe  factor  5  in  the  dividend,  which  is  the  required  quotient. 

31.  Divide  11  times  1476  by  6  times  132. 

41  Solution.  —  Striking   out  in  dividend 

to^  and  divisor  all  the  factors  common  to 

147^  y  TT  both,  there  remains  in  the  dividend  only 

— -  =  20  J  the  factor  41,  and  in  the  divisor  only  the 

^p^  X  p  factor  2.    Completing,  then,  the  division, 

Ar       ^  we  have  for  the  quotient  20^. 

32.  Divide  45  X  20  X  7  by  49  X  4  X  9. 

33.  (54  X  3  X  4  X  15)  -f-  (18  X  12  X  10)  =  ? 


FACTORS.  57 

86.    Rule  for  Cancellation. 

Strike  out  all  the  factors  common  to  both  dividend  and  di* 
visor,  and  then  divide  as  7nay  he  required. 

34.  Divide  255  x  63  x  4  by  340  X  12  X  7. 

35.  Divide  18  X  16  X  10  X  5  by  12  X  8  X  6  X  2. 

36.  Divide  50  X  36  X  14  by  54  X  10  X  4  X  3. 

37.  Divide  545  X  105  X  H  by  35  X  33  X  5. 

38.  Multiply  64  by  63  and  divide  the  product  by  168. 

39.  What  is  the  quotient  of  36  X  21  X  14  divided  by  2'^ 
X7x6? 

40.  How  many  tons  of  hay  at  $  24  a  ton  must  be  given  for 
4  cows  at  $  42  each  ? 

41.  How  many  yards  of  cloth  can  be  bought  for  $  95,  when 
24  yards  can  be  bought  for  $  120  ? 

42.  If  100  men  can  perfprm  a  piece  of  work  in  12  days,  in 
how  many  days  can  150  men  perform  it  ? 

43.  A  fort  has  pro^sions  for  225  men  12  months.  How 
long  will  it  last  675  men  ? 

44.  Exchanged  15  pieces  of  muslin,  each  containing  30 
yards  at  10  cents  a  yard,  for  3  pieces  of  flannel,  each  contain- 
ing 50  yards.     What  was  the  flannel  a  yard  ? 

GREATEST  COMMON  DIVISOR. 

45.  What  number  will  divide  both  21  and  35,  without  a 
remainder  ? 

46.  What  factor  is  common  to  21  and  35  ?     To  15  and  25  ? 

47.  What  is  the  greatest  number  that  will  divide  both  18 
and  30,  without  a  remainder  ? 

48.  What  is  the  greatest  factor  common  to  18  and  30? 
To  16  and  24.^ 

87.  A  Commoii  Divisor  of  two  or  more  numbers  is  any 
factor  found  in  each  of  them.     Thus, 

7  is  a  common  divisor  of  14  and  21. 


58  FACTORS. 

88.  The  Greatest  Common  Divisor  of  two  or  more  num- 
bers is  the  greatest  factor  found  in  each  of  them.     Thus, 

6  is  the  greatest  common  divisor  of  18  and  24. 

89.  Numbers  are  Prime  to  each  other  when  they  have 
no  common  factor  or  divisor.     Thus, 

9  and  14  are  prime  to  each  other. 

90.    Principle. 

The  greatest  common  divisor  of  two  or  more  numbers  is  the 
product  of  all  their  conimon  prime  factors, 

WRITTEN    EXERCISES. 

49.  What  is  the  greatest  common  divisor  of  84  and  132  ? 

2)  84,  132      ^^     84^=2x2x3x7  Solution.— 

2)42,    66         "*' 132:^2x2x3x11        ^^.    ^^^     *^^ 
^ prime    factors 

3)  21,    33  common  to   the 

7,     11  2  X  2  X  3  =  12,  Ans.  numbers  are  2,  2, 

and  3.  The  pro- 
duct of  these,  2  X  2  X  3  or  12,  is  the  greatest  common  divisor 
required. 

50.  What  is  the  greatest  common  divisor  of  36,  81,  135  ? 

51.  What  is  the  greatest  common  divisor  of  24,  42,  54,  and 
60? 

91.    Rule  for  finding  the  Greatest  Common  Divisor. 

Separate  the  numbers  into  their  prime  factors,  and  find  the 
product  of  all  such  as  are  common  to  the  numbers. 

What  is  the  greatest  common  divisor 

32.   Of  45  and  135  ?  56.    Of  20,  26,  and  38  ? 

53.  Of  90  and  105?  57.   Of  32,  48,  and  128  ? 

54.  Of  42  and  81  ?  58.    Of  45,  72,  and  81  ? 

65.   Of  132  and  156  ?  59.   Of  24,  51,  105,  and  729  ? 


FACTORS.  69 

60.  Having  three  rooms,  the  first  12  feet  wide,  the  second 
15  feet,  and  the  third  18  feet,  I  wish  to  purchase  a  roll  of  the 
widest  carpeting  that  will  exactly  fit  each  room  without  any 
cutting  as  to  width.     How  wide  must  it  be  ? 

92.  When  the  numbers  are  large,  or  cannot  readily  be 
separated  into  factors,  — 

Of  two  numbers  divide,  the  larger  hy  the  smaller,  and  the 
last  divisor  hy  the  last  remainder,  until  nothing  remains. 
The  final  divisor  is  the  greatest  common  divisor. 

If  more  than  two  numhers  are  given,  find  the  greatest  com- 
mon divisor  of  two  of  them,  then  of  this  divisor  and  a  third 
number,  and  so  on, 

61.  What  is  the  greatest  common  divisor  of  247  and  323  ? 

Solution.  —  If 

247)  323  (1  247    will    exactly 

247  divide  323,  it  will 

76)  247  (3  ^^^     *^^     greatest 

QOQ  common  divisor  of 

247   and   323.     It 

Greatest  common  divisor  =  19)  76  (4  ^^^^  ^^^  ^^^^^^  ^i. 

'^  vide  323,  for  there 

is  76  remainder. 
If  76  will  exactly  divide  247,  it  will  be  the  greatest  common  divisor 
of  76  and  247,  and  therefore  of  247  and  323.  It  will  not  exactly  di- 
vide 247,  for  there  is  19  remainder.  If  19  will  exactly  divide  76,  it 
will  be  the  greatest  common  divisor  of  19  and  76,  and  of  76  and  247, 
and  of  247  and  323.  19  will  exactly  divide  76,  and  therefore  it  is 
the  greatest  common  divisor  of  247  and  323. 

Find  the  greatest  commou  divisor  of 


62. 

336  and  480. 

67. 

2145  and  3471. 

63. 

925  and  1475. 

68. 

582  and  3724. 

64. 

308  and  506. 

69. 

10353  and  14877. 

65. 

172  and  1118. 

70. 

3528  and  4424. 

66. 

275  and  440. 

71. 

1764  and  2660. 

60  FACTORS. 

72.  744  and  906.      75.  2883  and  3131. 

73.  728  and  808.      76.  3178  and  3500. 

74.  756  and  1140.      77.  4872  and  9048. 

LEAST  COMMON  MULTIPLE. 

78.  Name  four  numbers  of  which  4  is  a  factor. 

79.  15  is  an  exact  number  of  times  what  two  numbers  ? 

80.  What  is  the  least  number  that  will  contain  both  3  and 
7  an  exact  number  of  times  ? 

81.  What  is  the  least  number  that  both  4  and  9  will  exactly 
divide  ? 

93.  A  Multiple  of  a  number  is  any  number  divisible  by 
it  (Art.  79,  note).     Thus, 

5,  10,  15,  etc.,  are  multiples  of  5. 

94.  A  Common  Multiple  of  two  or  more  numbers  is  any 
number  divisible  by  each  of  them.     Thus, 

48  is  a  common  multiple  of  4,  8,  and  12. 

95.  The  Least  Common  Multiple  of  two  or  more  numbers 
is  the  least  number  divisible  by  each  of  them.     Thus, 

24  is  the  least  common  multiple  of  4,  8,  and  12. 

96.    Principle. 

The  least  common  multiple  of  two  or  more  numbers  contains 
each  of  the  prime  factors  of  those  numbers^  but  7io  others. 

WRITTEN    EXERCISES. 

82.  What  is  the  least  common  multiple  of  21,  35,  and  45  ? 
21  =  3x7  „      3)21,35,45         Jl^'T:  7..  !t!!: 


rating    the    numbers 
into  their  prime  fac- 


35  =  5x7  ^""^   5)    7,35,15 

45  =  3x3x5  7^    3  tors,  we  find  the  pro- 

rr       Qi  r  ^^^^  ^^  ^^^  different 

3  X  3  X  5  X  i  =  31c»  factors,  using  each  the 


FACTORS.  61 

greatest  number  of  times  it  occurs  in  any  number,  to  be  3  X  3  X  5 
X  7,  or  315.  This  is  the  least  common  multiple,  because  it  contains 
each  prime  factor  of  the  numbers,  but  no  other  (Art.  96).  Or,  the 
different  prime  factors  may  be  found  by  the  second  process  annexed, 
the  two  methods  being  alike  in  principle. 

83.  What  is  the  least  common  multiple  of  7,  14,  15,  and 
21? 

84.  What  is  the  least  common  multiple  of  18,  28,  30,  and 
42? 

97.    Rule  for  finding  the  Least  Common  Multiple. 

Separate  the  numbers  into  their  prime  factors.  Take  the 
'product  of  all  the  different  fax^tors,  using  each  factor  the 
greatest  number  of  times  it  occurs  in  any  number. 

Note  1.  —  The  following  rule  is  sometimes  used. 

Strike  out  any  of  the  given  numbers  that  are  factors  of  any  of  the  others,  and 
divide  the  remaining  numbers  by  any  prime  factor  common  to  two  or  more  of 
them. 

Strike  out  from  the  resulting  quotients  and  undivided  numbers  all  that  are 
factors  of  any  of  the  rest^  and  divide  as  before. 

Thus  proceed  until  no  tioo  of  the  remaining  numbers  have  a  common  factor. 
The  product  of  the  divisors  and  remaining  numbers  will  be  the  least  common 
multiple  required. 

Note  2.  —  The  least  common  multiple  of  numbers  prime  to  each  other  is 
their  product. 

Find  the  least  common  multiple 

85.  Of  21,  33,  6Q.  89.   Of  8,  18,  24,  36. 

86.  Of  63,  72,  84.  90.   Of  7,  25,  12,  41. 

87.  Of  m,  88,  110.  91.   Of  28,  m,  100,  125. 

88.  Of  81,  63,  135.  92.   Of  24,  42,  54,  180. 

93.  What  is  the  least  common  multiple  of  24,  96,  100,  and 
144? 

94.  What  is  the  least  common  multiple  of  4,  11,  18,  20,  36, 
and  48  ? 

95.  What  is  the  least  sum  of  money  that  can  be  exactly  ex- 
pended for  sheep,  cows,  or  oxen,  at  $  5,  $  35,  and  $  50  each, 
respectively  ? 


62  FACTORS. 

MISCELLANEOUS    EXERCISES. 

96.  What  are  the  prime  factors  of  2520  ? 

97.  How  many  times  are  2  and  3  respectively  factors  in 
5760? 

98.  What  is  the  greatest  factor  common  to  689  and  1573  ? 

99.  rind  the  sum  of  all  the  prime  numbers  between  70  and 
100. 

100.  Employ  cancellation  in  dividing  14  X  15  X  16  X  24 
X  48  X  60  by  7  X  30  X  8  X  8  X  6  X  12  X  3. 

101.  How  many  times  is  the  greatest  common  divisor  of  48, 
36,  72,  24  contained  in  their  least  common  multiple  ? 

102.  Find  the  sum  of  the  composite  numbers  between  100 
and  120,  inclusive. 

103.  What  is  the  difference  between  the  greatest  common 
divisor  and  the  least  common  multiple  of  160,  352,  and  992  ? 

104.  Divide  1008  by  168,  using  prime  factors  and  cancella- 
tion. 

105.  When  hay  is  $  24  a  ton,  how  many  barrels  of  flour,  at 
$  8,  will  exactly  pay  for  35  tons  of  hay  ? 

QUESTIONS. 

78.  What  are  the  factors  of  a  number  ?  79.  What  is  a  composite 
number  1     80.   What  is  a  prime  number  1 

81.  What  is  a  prime  factor  1  83.  How  do  you  find  the  prime  fac- 
tors of  a  number  ? 

84.  What  is  cancellation  1  85.  What  are  principles  of  cancella- 
tion 1    86.   How  is  cancellation  performed  ? 

87.  What  is  a  common  divisor  of  two  or  more  numbers  ?  88.  The 
greatest  common  divisor  of  two  or  more  numbers  ?  90.  What  is  a 
principle  of  common  divisors?  91.  How  is  the  greatest  common 
divisor  found  ? 

93.  What  is  a  multiple  of  a  number  1  94.  A  common  multiple  of 
two  or  more  numbers  ?  95.  The  least  common  multiple  of  two  or 
more  numbers  ?  96.  A  principle  of  multiples  ?  97.  How  is  the 
least  common  multiple  found  ? 


COMMON   FRACTIONS.  .  63 


COMMON    FRACTIONS. 

98.  1.  If  a  unit,  as  an  apple,  is  divided  into  two  equal 
pieces,  what  part  of  the  whole  will  one  piece  be  ? 

2.  If  an  apple  is  divided  into  three  equal  pieces,  what  part 
of  the  whole  will  one  piece  be  ?     Two  pieces  ? 

3.  If  a  single  thing  is  divided  into  four  equal  pieces,  what 
part  of  the  whole  will  one  piece  be  ?  Two  pieces  ?  Three 
pieces  ? 

4.  How  many  halves  in  an  apple  ?  How  many  thirds  ? 
How  many  fourths  ? 

5.  What  is  meant  by  one  half  of  a  unit  ?  By  one  third  ? 
By  two  thirds  ?     By  one  fourth  ?     By  three  fourths  ? 

6.  Which  are  the  larger  parts  of  an  apple,  halves  or  thirds  ? 
Thirds  or  fourths  ? 

99.  A  Fraction  is  one  or  more  of  the  equal  parts  of  a 
unit. 

The  Unit  of  the  Fraction  is  the  unit  divided,  and  a 
Fractional  Unit  is  one  of  the  equal  parts  into  which  it  is 
divided. 

100.  The  Denominator  of  a  fraction  is  the  number  that 
shows  into  how  many  equal  parts  the  unit  is  divided. 
Thus, 

Three  is  the  denominator  of  two  thirds. 

101.  The  Numerator  of  a  fraction  is  the  number  that 
shows  how  many  of  the  equal  parts  of  the  unit  are  taken. 
Thus, 

Two  is  the  numerator  of  two  thirds. 

102.  The  Terms  of  a  fraction  are  its  numerator  and 
denominator.     Thus, 

2  and  3  are  the  terms  of  the  fraction  |. 


64  .  COMMON   FEAGTIONS. 

103.  A  Common  Fraction  is  a  fraction  expressed  by- 
writing  the  numerator  above,  and  the  denominator  below, 
a  dividing  line.     Thus, 

Three  fourths  of  a  dollar  is  written  $f,  3  being  the 
numerator,  4  the  denominator,  1  dollar  the  unit  of  the 
fraction,  and  \  dollar  t\\Q  fractional  unit 

104.  An  Integer  may  be  expressed  in  a  fractional  form, 
by  writing  1  under  it  for  a  denominator.     Thus, 

2.  may  be  written  |,  and  read  2  ones ;  7  may  be  written 
^,  and  read  7  ones ;  etc. 

105.  A  Proper  Fraction  is  a  fraction  whose  numerator 
is  less  than  its  denominator.     Thus, 

^  and  Y^-j  are  proper  fractions. 

106.  An  Improper  Fraction  is  a  fraction  whose  numer- 
ator is  not  less  than  its  denominator.     Thus, 

1^  and  -l|  are  improper  fractions. 

107.  A  Mixed  Number  is  an  integer  and  a  fraction 
united.     Thus, 

3J,  read  three  and  one  fourth,  is  a  mixed  number. 

108.  A  Fraction  may  be  regarded  as  an  indicated  divis- 
ion (Art.  63),  the  numerator  being  the  dividend,  and  the 
denominator  the  divisor.     Thus, 

I  of  1  inch  is  the  same  as  ^  of  3  inches,  or  3  inches 
divided  by  4. 

109.  The  Value  of  a  fraction  is  the  quotient  of  the  nu- 
merator divided  by  the  denominator. 

EXERCISES. 

Express  in  figures  : 

7.  Three  sevenths.  9.   Nine  sixteenths. 

8.  Seven  elevenths.  10.   Seventeen  ones. 


COMMON   FRACTIONS.  65 

11.  One  twenty-first.  14.  Three  and  three  twenty-ninths. 

12.  Eleven  thirty-seconds.    15.  Twenty-three  and  three  fifths. 

13.  Nineteen  fortieths.         16.  Eight  and  nine  twelfths. 


REDUCTIOlSr  OF  FRACTIONS. 

ORAL    EXERCISES. 

17.  In  J  of  an  apple  how  many  fourths  of  an  apple  ?   how 
many  eighths  ? 

18.  In  §  of  an  apple  how  many  sixths  ?  how  many  ninths  ? 

19.  Name  a  fraction   equal  to  ^.     Name  a  fraction  equal 
to  f. 

20.  Express  f  in   terms  2  times   as   large.      3  times  as 
large. 

21.  Change  f  to  t%,  f  to  j%,  J  to  j%. 

22.  How  is  the  fraction  f  changed  to  twelfths  ? 

23.  How  many  halves  are  there  in  |  ?     How  many  thirds 
are  there  in  j^^  ? 

24.  How  many  tenths  of  a  melon  in  ^g  ?  how  many  fifths  ? 

25.  Express  f  in  larger  terms ;  f  in  larger  terms ;  -^  in 
smaller  terms ;  |f  in  smaller  terms. 

110.    Reduction  of  Fractions  is  changing  their  form  with- 
out changing  their  value. 

To  Larger  Terms. 

26.  Change  |  to  sixteenths. 

Solu  Hon.  —  The  required  den  om- 
inator  is  4  times  the  given  denom- 
inator.   Multiplying  both  terms  of 
the  fraction  by  4  gives  ^. 
1"  X  4        ^^         "if  ^^   ^^^1  ^^   observed  from  the 

illustration,  that  f  of  the  square 
equals  }|  of-  it,  the  multiplication  increasing  the  number  of  fractional 
uuits  4  times,  and  making  each  one  fourth  as  large,  so  that  the  valuer 
of  the  fraction  is  not  changed. 

6 


1 

F 

■ 

if 

: 

Si 



66  COMMON   FRACTIONS. 

111.     Principle. 

Multiplying  both  terms  of  a  fraction  by  the  same  number 
does  not  change  its  value. 

27.  Change  to  12ths  ^  I,  i,  J,  §,  |,  |. 

28.  Change  to  18ths  i,  J,  ^  ^,  f,  f,  |,  J,  |,  |. 

29.  Change  to  20ths  J,  i,  i,  |,  ^^,  |,  ^^,  ^^,  ^^. 

30.  Change  to  24ths  i,  J,  i  J,  i,  ^i,,  f,  |,  |,  |,  |,  |,  /^,  |^. 

31.  Change  to  36ths  f,  f ,  #,  f ,  J,  \^,  i,  f ,  ^^,  /^. 

32.  Change  to  48ths  %,  f,  i,  f,  |,  ^^^^  ^J,  ,3^,  ^^,  i  j,  J|. 

WRITTEN    EXERCISES. 

33.  Change  |  to  twenty-fourths. 

Solution.  —  To  change  8ths  to  24ths, 

•|  =  1^1  =  2^^  we  must  multiply  both  terms  of  the 

Qp  fraction  by  3.    Doing  this,  we  obtain 

3  ■^.     Or, 

"g"  of  1  =  5;  of  2^4  =  2"T  Since  1  is  24  twenty-fourths,  \  of 

1   is  ^  of  24  twenty-fourths,   or   3 
twenty-fourths,  and  f  of  1  is  3  times  3  twenty-fourths,  or  ^. 

34.  Change  f  to  seventy-fifths. 

112.  To  change  a  fraction  to  larger  terms  : 

Rule. 

Divide  the  required  denominator  by  the  given  denominator^ 
and  multiply  both  terms  of  the  fraction  by  the  quotient. 

Change : 

35.  ^  to  84ths.  39;  f   to  168ths. 

36.  I  to  32ds.  40.  if  to  189ths. 

37.  J  to  54th s.  41.  I   to  48ths. 

38.  \l  to  196ths.  42.     f  to  576ths. 

To  Smaller  Terms. 

113.  A  fraction  is  in  its  Smallest  Terms  when  its  terms 
have  no  common  factor. 


COMMON   FRACTIONS. 


67 


ORAL   EXERCISES. 


43.   Change  |f  to  smallest  terms. 


Solution.  —  Dividing  both  terms 

of  ^f  by  4,  we  have  |,  whose  terms 

have  no  common  factor ;  hence, 

J[-|  changed  to  smallest  terms  is  f , 

It  will  be  observed  from  the 

illustration  that  ^|  of  the  square 

equals  f  of  it,  the  division  increasing  the  size  of  the  fractional  units  4 

times,  while  their  number  is  a  fourth  as  large,  so  that  the  value  of  the 

fraction  is  not  changed. 

114.     Principle. 

Dividing  both  terms  of  a  fraction  by  the  same  number  does 
not  change  its  value. 

Change  to  smallest  terms ; 

44.  i,  h  h  I.  T%  A,  A.  \%  A,  t\. 

45.  f,  f,  S;  A,  Hv  A,  H,  H,  A;  M. 

46.  A,  A,  \h  Ih  hh  \h  iT,  ^T,  hh  «. 

47.  A.  if;  li  Ih  hh  If,  ih  IS,  if;  if • 

48.  ft,  ft,  A;  II;  ft;  f I;  il;  ft;  A;  %h 

49.  A;  if;  if;  IS;  If;  M;  If;  ft;  if;  f«- 

50.  i«,  fi,  H,  f §,  ii  f f,  f  J,  Jt,  A;  f |. 

51.  A%;  A%;  A%;  A^o;  A%;  A%;  A%;  A^o;  A%;  AV 


WRITTEN    EXERCISES. 

52.    Change  f  §  to  smallest  terms. 

Solution.  —  Dividing  both  terms  of  -f -J  by  3, 
we  have  J-|,  and  dividing  both  terms  of  \^  by 
5,  we  have  -f,  whose  terms  have  no  common 
factor  ;  hence,  |f  changed  to   smallest  terms 


3)3  0.  _  10. 
3)75   ~   2  5 


5)  10.   —  1 
5)25    -"    5 


68  COMMON   FRACTIONS. 

53.  Change  j^^  to  its  smallest  terms. 

54.  Change  Iff  to  its  smallest  terms. 

115.    To  change  a  fraction  to  its  smallest  terms : 

Rule. 

Divide  both  terms  of  the  fraction  by  any  common  divisor , 
treat  the  resulting  fraction  in  the  same  way,  and  so  continue 
until  a  fraction  is  found  whose  terms  are  jjrime  to  each  other. 

Note.  —  The  greater  the  common  divisor  used,  the  shorter  will  be  the  pro. 
cess. 

Change  to  their  smallest  terms  : 


55. 

m- 

59. 

m- 

63. 

m- 

67. 

1^7^- 

56. 

i^- 

60. 

m- 

64. 

m- 

68. 

im- 

57. 

t'A- 

61. 

m- 

65. 

^^Vn- 

69. 

m- 

58. 

Iff- 

62. 

in- 

66. 

VsV- 

70. 

mh 

An  Integer  or  Mixed  Number  to  an  Improper  Fraction. 
ORAL   EXERCISES. 

71.  In   2   apples   how  many  fourths  of   an  apple  ?     In   5 
apples  ?     In  8  apples  ? 

72.  Eeduce  6  to  fourths  ;  3  to  fifths  ;  7  to  sixths. 

73.  How  many  fourths  of  an  apple  in  3J  apples  ?     Iw  6J 
apples. 

Change  to  improper  fractions : 

74.  1§,  If,  li,  1^,  1|,  2f,  3^,  5J,  4i,  3§. 

75.  2t,  25,  3i,  3^,  3f,  3f,  3J,  4^,  4§,  4f. 

76.  4i,  4f,  4J,  51,  5/^,  5?,  5i,  5|,  5|,  GJ. 

77.  6§,  6f,  65,  6f,  65,  6|,  7i-,  8^,  8§,  8|-. 

78.  81  9J,  85,  4^j,  6^„  4f,  6/^,  8^,  9^,  5f 


COMMON    FRACTIONS.  69 

WRITTEN    EXERCISES. 

79.  Change  19  to  eighths. 

19 

8  eighths.  Solution.  —  In  1  there  are  8  eighths ;  in  19 

152  there  are  19  times  8  eighths,  or  i|-^. 

8 

80.  Change  197  to  the  form  of  a  fraction. 

81.  Change  21|  to  ninths. 

2lf  Solution.  —  In  1  there  are  9  ninths,  and  in  21  there 

zr-i  are  21  times  9  ninths,  or  189  ninths  :  189  ninths  and 

194 

"sT  5  ninths  are  194  ninths,  or  J-|4-. 

82.  Change  27i^5  to  thirteenths. 

83.  Change  37^^  to  elevenths. 

116.  To  change  an  integer  or  mixed  number  to  an  im- 
proper fraction : 

Rule. 

Multiply  the  integer  by  the  denominatoi',  and,  if  there  is  a 
fractional  part,  add  its  numerator  to  the  product,  and  write 
the  result  over  the  denominator. 

84.  Eeduce  93  to  fifteenths ;  107  to  twentj^-firsts. 

85.  Reduce  115  to  a  fraction  whose  denominator  is  13. 

Keduce  to  fractions  in  their  smallest  terms : 

86.  9-^^.  90.    14|§.  94.    81^^. 

87.  56t\.  91.    98|f.  95.    25/7J5. 

88.  104^.  92.   3|f.  96.   153t^. 

89.  4tJ^.  93.   142f.  97.   ll^^U^, 

An  Improper  Fraction  to  an  Integer  or  Mixed  Number. 
ORAL   EXERCISES. 

98.  How  many  dollars  in  32  quarter-dollars  ?  In  40  quarter- 
dollars  ?     In  $  -y  ?     In  $  4^-  ?     In  $  ^^  ? 


70  COMMON   FRACTIONS, 

Change  to  integers  or  mixed  numbers  : 

99.  f ,  ¥,  -¥-,  ¥,  h  h  ¥,  -VS  ¥.  ¥• 
100.  -\%  V-;  f;  h  -¥-;  V-,  -VS  -VS  ¥;  ¥-. 
lOL  -V-,  -2^s  ¥;  ¥,  ¥.  ¥-,  ¥-,  ¥.  -¥-.  -¥- 

102.  i^-,  -3/-,  -Y-,  -%  ¥;  -¥;  ¥;  f  f ;  f  i;  -¥- 

103.  ^-i-,  VS  ¥;  v.  ¥;  n,  ¥.  tf;  tl;  ¥- 

WRITTEN    EXERCISES. 

104.  Change  ^^^-  to  an  integer  or  mixed  number. 

19)  793  Solution.  —  As  19  nineteenths  are  1,  793  nine- 

-^  teenths  will  be  as  many  ones  as  times  19  in  793, 

19  or  41^|.     Ans.  41i|. 

14 

105.  Change  -\-^  to  an  integer  or  mixed  number. 

106.  Change  ^yf  ^  to  an  integer  or  mixed  number. 

117.    To  change  an  improper  fraction  to  an  integer  or 
mixed  number : 

Rule. 
Divide  the  numerator  hy  the  denominator, 

Eeduce  to  integers  or  mixed  numbers : 

107.  ff.  112.  IJ,  117.  ^V-. 

108.  ^J^.  113.  -Vt^..  118.  ^^K 

109.  ^^^,  114.  lf&.  119.  m-' 

110.  fg-J.  115.  ifp.  120.  -^ili^. 

Ill  le.  116.  -Wjf-.         121.  ^n^. 

Fractions  to  Fractions  having  a  Common  Denominator. 
ORAL  EXERCISES. 

122.  Express  J  as  tenths ;  J  as  tenths. 

123.  Express  J  and  \  each  as  twelfths. 


COMMON   FRACTIONS.  71 

124.  Change  f  and  |  to  fractions  having  the  same  denomi- 
nator. 

125.  Change  -f  and  f  to  fractions  having  the  same  denom- 
inator.    What  is  a  multiple  of  6  and  5  ? 

126.  Change  ^,  §,  and  J  to  fractions  having  the  same  de 
nominator. 

127.  Express  f,  |,  and  /^  each  as  fortieths. 

128.  Express  §,  f,  and  f  each  as  eighteenths. 

129.  What  is  the  least  common  multiple  of  3,  6,  and  9  ? 

118.  Fractions  have  a  Common  Denominator  when  their 
denominators  are  alike. 

119.  The  smallest  denominator  common  to  two  or  more 
fractions  is  their  Least  Common  Denominator. 

120.  A  Common  Denominator  of  several  fractions  must 
be  some  common  multiple  of  their  denominators. 

The  Least  Common  Denominator  of  several  fractions  must 
be  the  Least  Common  Multi;ple  of  their  denominators. 

WRITTEN     EXERCISES. 

130.  Change  f  and  f  to  fractions  having  a  common  denom- 
inator. 

2X7  _  14  Solution.  —  As   the   product  of  the   denomina- 


tors,  5x7,  or  35,  is  their  common  multiple,  35 


5X7  —  3  5 

3.x  5  __  15.       must  he  a  common  denominator  of  the  fractions. 

Changing   then  the   fractions  to  thirty-fifths,  we 
have  \\  and  f|. 

131.  Change  f  and  |  to  fractions  having  a  common  denom- 
inator. 

132.  Change  §,  |,  and  -g  to  fractions  having  the  least  com- 
mon denominator. 


72  COMMON  FRACTIONS. 


4 
^  01    12    —    12 


Solution.  —  The  least  common  multiple  of 
the  denominators  is  12,  which  must  be  the  least 

3      „  ^  _9^       common  denominator  of  the  fractions.     Chang- 

^  ot   1  2   -  12        ij^g  thentt-  ---—  - -  « 

2 
5   ^ir  >^  _  1_0       T^^,  and  ^0. 
^  ot  1  2  —  1 2 


^        ^  ^         ^  ^       ing  then  the  fractions  to  tweKths,  we  have  ^jy 


133.  Change  |,  f ,  and  |  to  fractions  having  the  least  com-^ 
mon  denominator. 

134.  Reduce  |,  |,  and  \^  to  fractions  having  the  least  com- 
mon denominator. 

121.  To  reduce  to  fractions  having  the  least  common 
denominator : 

Rule. 

Change  each  fraction  to  its  smallest  terms.  Divide  the 
least  common  multiple  of  the  denominators  hy  the  denomina- 
tor of  each  fraction,  and  multiply  both  terms  of  the  fraction 
hy  the  quotients 

Note.  —  When  the  denominators  are  nmtnally  prime,  take  their  product  for 
the  common  denominator,  and  multiply  each  numerator  by  all  the  denomina- 
tors except  its  own  for  the  new  numerator. 

Reduce  to  fractions  having  the  least  common  denomina- 
tor: 

135.  f  and  ^^.  141.  ^^,  \%  and  \\. 

136.  f  and  -^.  142.  ^-^   \,   and  -V-. 

137.  T^^andH.  143.  ^,  Jf,  ^^,  and  ^^. 

138.  /^  and  ^^,  144.  ^7^,  yj^,  y^^^,  and  7. 
139-  -ft,  ifr;  and  ^^.  145.  ^^,  f ,  ^,  and  /^. 
140.  J,  $,  andif  146.  |, -,i>^,  and  |. 

147.  Reduce  ^,  f ,  and  -^  to  fractions  having  the  least  com- 
mon denominator,  and  show  by  the  numerators  which  of  the 
fractions  has  the  greatest  value. 


COMMON  FRACTIONS.  73 

ADDITIOISr. 

ORAL   EXERCISES. 

148.  Mary  paid  |  of  a  dollar  for  a  book,  |  of  a  dollar  for  a 
hat,  and  f  of  a  dollar  for  a  handkerchief.  How  many  eighths 
of  a  dollar  did  she  spend  for  all  ? 

149.  Smith  owns  ^  of  a  vessel  and  Keen  f .  How  many 
sixteenths  does  Keen  own  ?  What  part  of  the  vessel  do  both 
own  ? 

Give  the  sum  of  the  following  fractions : 


150. 

151. 

152. 

153. 

154. 

h  +  i 

i  +  1 

f   +  i 

t^  +  tV 

i+i+   i 

§   +  1 

t  +  § 

i   +  h 

1  +  i 

4  +  i+  i 

h  +  i 

i  +  f 

1   +  « 

tV+   f 

i  +  i  +  ^ 

3  +  i 

f  +  4 

tV+  i 

i  +  t 

§  +  1  +   1^2 

A+  4 

1  +  i 

f  +  4 

A+  § 

l  +  f+  f 

122. 

Like  Fractions  are  like 

parts  of 

the 

same  unit. 

Thus, 

f  ofa 

dollar  and 

|-  of  a  dollar  • 

also,  i\ 

and 

^■"j-  are  like 

fractions 

123.    Principle. 

Only  ^ 

like  fradioTis  can  he  added. 

WRITTEN    EXERCISES. 

155.  What  is  the  sum  of  f ,  %,  and  {^  ? 

5      ,      3     _^  11  _  Solution.  —  Reducing  the  given    frac^ 

^           ^          ^^  tions  to  fractions  having  the  least  comnion 

If  +  Ti  +  If  =  denominator,   we   have   |f,    ^%   and  ||, 

||.  -^  22 T  =  2|-  which  added  give  f-J-  =  2^,  or  2-J. 

156.  What  is  the  sum  of  f ,  H>  ^^^  tS  ^ 

157.  What  is  the  sum  of  ^,  \i,  and  ^  ? 


74  COMMON   FKACTIONS. 

124.    Rule  for  Addition  of  Fractions. 

Change  the  fractions^  if  necessary,  to  fractions  having  a 
common  denominator,  add  the  numerators,  write  the  sum  over 
the  common  denominator,  and  simplify  the  result,  if  needfuL 

158.  Add  I,  -i^,  and  -i^,  162.  Add  -f^,  /y,  and  ^^. 

159.  Add  if,  11,  and  l{,  163.  Add  %S  h  and  f  J. 

160.  Add  If,  and  f^.  164.  Add  J,  ^^,  IJ,  and  /^. 

161.  Add  f ,  ^j,  and^.  165.  Add  ^3_,  ||,  zj^  and  AV- 

166.  What  is  the  value  of  f  +  If  +  V  +  i  ^ 

125.  When  there  are  mixed  numbers,  the  fractions  and 
the  integers  may  he  added  separately y  and  the  results  united, 

167.  What  is  the  sum  of  15|  and  24f  ? 

168.  What  is  the  value  of  37|  -f  109f  +  341^t-  ? 

169.  A  man  paid  $  4|  for  a  hat,  $  16  J  for  a  coat,  and  $  5^ 
for  a  pair  of  boots.     How  much  did  he  pay  for  the  whole  ? 

170.  A  has  in  his  farm  160|  acres,  B  has  67 1%  acres,  and 
C  has  85^^  acres.  What  is  the  number  of  acres  in  the  three 
farms  ? 

SUBTRACTION. 
ORAL   EXERCISES. 

171.  A  man  owned  \^  of  a  ship  and  sold  /^.  What  part  of 
the  ship  had  he  left  ? 

172.  \^  less  -^^  is  what  part  of  1  ? 

3  73.   How  much  is  \l  less  |J  ?     ^f  less  \\  ? 
174.    How  much  is  ||  less  f  ? 

Solution.  —  f  is  il ;  and  f|  less  ^f  is  ^. 


175. 

176. 

177. 

§  -i  =  ? 

*  -  A  =  ? 

3-  1  =? 

4  -i  =  ? 

i  -  f  =  ? 

^-Tft=? 

T^  -  i  =  ? 

8-  i  =? 

i  -    i    =  ? 

#  -4  =  ? 

S-  i  =? 

S-  f  =? 

A-i  =  ? 

1-  §  =? 

«-  i=? 

COMMON  FRACTIONS.  75 

178.  179. 

f  -  T^  -  ?  #  -  A  =  ? 

I  -  A  =  ?  f  -  t't  =  ? 

!--«-?  il  -  «  =  ? 

180.  Jane  had  $  2  J,  and  spent  $  1|.  How  much  had  she 
left  ? 

126.     Principle. 

Only  like  fractions  can  he  subtracted  the  one  from  the 
other. 

WRITTEN    EXERCISES. 

181.  From  if  subtract  /_. 

13 ^7_  __  3.1  _  18.     -^  11  Solution.  —  Reducing  the 

16  12—   4848?  or   48  .  ^        .  „    ^  . 

given  iractions  to  iractiona 

having  the  least  common  denominator,  we  have  ff  and  |f.     |-|  sub- 
tracted from  II  leaves  W. 

182.  Find  the  difference  between  fj  and  f  J. 

183.  From  §g  take  §g.  184.    From  i  J  take  ^\. 

127.     Rule  for  Subtraction  of  Fractions, 

Change  the  fractions,  if  necessary,  to  fractions  having  a 
common  denominator,  find  the  difference  of  the  numerators, 
and  write  it  over  the  common  denominator. 

185.  From  \l  take  ^^,  189.  From  f -f  take  f  |. 

186.  From  JJ  take  ■^^.  190.  From  ^V^  *ake  yVo- 

187.  From  §J-  take  ^^.  191.  From  ^^  take  yi§^. 

188.  From  \{\  take  ^j^^.  192.  From  %\  take  |f . 

193.  What  is  the  difference  between  ^f  and  -^  ? 

194.  I  copied  by  mistake  Jf  instead  of  ^^.  What  is  the 
amount  of  error  I  made  ? 

195.  How  much  is  49f  less  31f  ? 


76 


COMMON  FRACTIONS. 


49f  =  49H 
31| 


=  4811 

=  3l|i 

1711 


Solution.  —  Changing  the  fractiuns  to 
fractions  having  a  common  denominator, 
we  have  49 Jf  and  31f|-.  As  we  cannot 
take  1^  from  ^,  we  take  1,  or  |f,  from 
49,  leaving  48  ;  and  adding  the  fl  to  |-|, 

we  have  48|-|.     Subtracting  ||  from  ||,  and  31  from  48,  we  have, 

as  the  result  required,  17|-|-. 

196.  31|  -  101  r=  ?  199.   2911  -16'l  =  ? 

197.  63  -  54/2  =  ?  200.   311  -  30i|  =  ? 

198.  73^^-67jIj,  =  ?  201.   103i-gf-99H=.? 

202.  A  boy  paid  for  a  sled  $  7|,  and  for  skates  $  If.  How 
much  more  did  he  pay  for  the  sled  than  for  the  skates?      > 

203.  The  greater  of  two  numbers  is  150,  and  the  smaller  is 
147iJ.     What  is  their  difference  ? 

204.  Two  men  start  from  the  same  place  and  travel  in  the 
same  direction.  When  one  has  gone  17  2^0  miles  and  the  other 
19  f  miles,  how  far  are  they  apart  ? 


MULTIPLICATION. 

A  Fraction  multiplied  by  an  Integer. 
ORAL    EXERCISES. 

205.    Multiply  j\  by  4. 


j 

— 

■— 

E 


3_  X   4     _        12  ^^^,  ;;^  _  a 

16  —         16?  ^^  16:4—  'i 

Solution.  —  Multiplying  the  numerator  of  ^^  by  4  gives  ^|,  equal 
to  I ;  or,  dividing  the  denominator  of  ^-^  by  4,  gives  |.  In  either 
case,  ^  is  multiplied  by  4.  For  it  will  be  seen  by  the  illustration 
that  multiplying  the  numerator  by  4  increases  the  number  of  fractional 
units  4  times,  while  their  size  remains  unchanged  ;  and  that  dividing 
the  denominator  hy  4  increases  the  size  of  the  fractional  units  4  times, 
wliile  their  number  is  unchanged.  The  eflfect  of  either  process, 
therefore,  is  to  increase  the  value  of  the  fraction  4  times. 


COMMON  FRACTIONS.  77 

206.  At  $J  each,  how  many  dollars  will  12  arithmetics 
cost  ? 

207.  At  the  rate  of  f  of  a  bushel  of  grain  a  week,  how  many 
bushels  of  grain  will  a  horse  consume  in  6  weeks  ? 

208.  If  1  hat  costs  $  3 J,  what  will  8  hats  cost  ? 

209.  Multiply  the  fractions  in  Exercises  27  to  32  by  3 ;  b}; 
4;  by  5;  by  6;  by  7;  by  8;  by  9;  by  10;  by  12. 

210.  Multiply  the  mixed  numbers  in  Exercises  74  to  78  by 
4;  by  5;  by  6;  by  8;  by  9;  by  10;  by  12. 

128.     Principle. 

A  fraction  is  multiplied  either  by  multiplying  its  numerator 
9r  by  dividing  its  denominator, 

7 

/ 

WRITTEN    EXERCISES. 

211.  Multiply  \\  by  7. 

Solution, 

Or, 

212.  Multiply  ^y  by  6.  216.  Multiply  ^  by  35._^ 

213.  Multiply  ig-  by  16.  217.  Multiply  ^^\  by  18. 

214.  Multiply  §f  by  40.  218.  Multiply  |J  by  48. 

215.  Multiply  II  by  85.  219.  Multiply  -W-  by  76. 

220.    What  is  the  product  of  63|  by  5  ? 

03  3  Solution.  —  63f    is   63 

1X5==  "V"  =     3|  5  and  |.    I  multiplied  by  5 

^\~\§  gives  3|,  and  63  multi- 

^^'  — ^  plied    by    5    gives    315. 

63  X  5  =315  ^1"  These  results  added  give, 

Q-|o3  as  the  product  required, 

318^  31 8|.  _   „ 


78  COMMON  FRACTIONS. 

221.  Multiply  40t7^  by  27.         223.   Multiply  39^^^  by  75. 

222.  Multiply  81f  by  63.  224.    Multiply  V25j\  by  80. 

225.  What  will  72  barrels  of  St.  Louis  flour  cost  at  $  8f  a 
barrel  ? 

An  Integer  mulfiplied  by  a  Fraction. 

ORAL     EXERCISES. 

226.  If  a  bushel  of  corn  costs  64  cents,  what  will  J  of  a 
bushel  cost  ?  J  of  a  bushel  ? 

227.  If  a  pound  of  tea  costs  60  cents,  what  will  J  of  a  pound 
cost  ?  f  of  a  pound  ? 

228.  What  is  f  of  4  ? 

Solution.  —  ^  of  4  is  4",  and  -f  of  4  must  be  5  times  f ,  or  ^',  equal 
to  2f . 

229.  What  is  t3^  of  8  ?  f  of  3  ?  f  of  5  ?  |  of  10  ? 

230.  What  is  the  cost  of  J  of  a  cord  of  wood  at  $  6  a  cord  ? 
What  is  $  6  multiplied  by  J  ? 

129.     Principle. 

Multiplying  a  number  by  a  fraction  is  taking  such  a  part 
yf  the  number  as  is  denoted  by  the  fraction. 

WRITTEN    EXERCISES. 

231.  Multiply  180  by  ■^, 

15)  180  180  Solution.  — 180  x  j\  is  the  same 

"12      ^  _^  as  ^4^  of  180.   ^V  0^  180  =  12,  and 

4      ^^'     15)  720  t\  of  180  =  4  times  12,  or  48. 

48  48  Or,  as  ^  is  the  same  as  -^  of 

4,  180  X  Y^  is  the  same  as  ^  of 

i  times  180.     4  times  180  =  720,  and  ^  of  720  =  48. 

232.  Multiply  108  by  J.  236.  Multiply  144  by  |J. 

233.  Multiply  89  by  y\.  237.  Multiply  239  by  -^ir- 

234.  Multiply  375  by  |J  238.  Multiply  404  by  y\. 

235.  Multiply  137  by  f  239.  Multiply  376  by  |f 


COMMON   FRACTIONS. 


79 


240.   Multiply  33  by  3f. 


33  X  f  -  13i 
33  X  3  r=  99 


Or 


33 

99 


241. 
242. 
245. 


112i 

1121 

Multiply  110  by  9|. 
Multiply  85  by  13j. 


Solution.  —  3|  is  3  and  | ; 
33  multiplied  by  |  gives  13|, 
and  33  multiplied  by  3  gives 
99.  These  results  added 
give,  as  the  product  required; 
112f 

243.  Multiply  84  by  9t^3^. 

244.  Multiply  145  by  llf 


What  cost  25^  acres  of  land  at  $  50  an  acre  ? 


A  Fraction  multiplied  by  a  Fraction. 
ORAL    EXERCISES. 

246.  What  part  of  a  dollar  is  J  of  J  of  a  dollar  ?  J  of  J  of  a 
dollar  ? 

247.  What  part  of  a  dollar  is  J  of  §  of  a  dollar  ?  J  of  |  of  a 
dollar  ? 

248.  What  is  i  of  i  ?  ^  of  ^  ?  i  of  ^  ?  i  of  ^  ? 

249.  What  is  J  of  f  ?  i  of  f  ?  J  of  f  ?  I  of  f  J  ?     ^ 

250.  How  much  is  J  of  f  ? 


1    nf    1 


1  6' 


1    nf    3    _      3 


Solution.  —  ^  of  f  is  ^^,  and  f  of  |  is  3  times   j\,  equal  to 
This,  also,  appears  by  the  illustration. 


A- 


251. 

§   X|  =  ? 

i  x|  =  ? 

i  X  +  -? 
tVX|  =  ? 

Ax«  =  ? 


252. 

i  xf  =  ? 
■A  X  f  =  ? 
A  x  f  =  ? 
Ax^  =  ? 


253. 

tV  X  i  =  ? 
f  xi  =  ? 
I  xf  =  ? 
?  x#  =  ? 

T^^X|  =  ? 


254.    Give  the  product  of  each  pair  of  fractions  in  Exercises 
150  to  153. 


80  COMMON   FRACTIONS. 

130.  A  Comjpound  Fraction  is  a  fraction  of  a  fraction, 
as  I  of  |,  and  may  be  considered  an  expression  of  multi- 
plication. 

A  Simple  Fraction  is  a  fraction  not  connected  with 
another,  and  having  both  its  terms  integers. 

WRITTEN    EXERCISES. 

255.  Multiply  f  f  by  f . 

g  Solution.  —  f -J 

3  0  v/  5  _  15  0  _  2  5     oj.    ,S^x  5  _  2  5  multiplied   by  I 

SlXe  —  186~3  1'"^'    31X^6-—  31  .       .^  "^^ 

IS  the  same  as  f 

of  f  ? ;  i  ol'  If  is  A'e .  and  |  of  f  f  is  5  times  ^^^  equal  to  ^|i  or  f  f . 
Note. —  By  canceUation  we  may  shorten  the  process,  as  shown  in  the  second 
form  of  the  work. 

256.  Multiply  If  by  \\ ;  Jf  by  f  f . 

131.     Rule  for  Multiplication  of  Fractions. 

Write  the  'product  of  the  numerators  over  the  product  of  the 
denominators. 

Note.  —  The  rule  is  general,  since  a  mixed  number  may  be  changed  to  an 
improper  fraction,  and  an  integer  may  be  expressed  in  a  fractional  form. 

257.  Multiply  f  f  by  27.         260.    Multiply  J  by  f  of  ^. 

258.  Multiply  tf  by  e.  261.    Multiply -Y- by  |  of  t%. 

259.  Multiply  f  gj  by  f .         262.    Multiply  f f  by  \  of  f i. 

263.  What  is  the  value  of  f  of  f  of  ^  ? 

264.  What  is  the  value  of|X|XtXf? 

265.  What  cost  llf  cords  of  wood  at  $  7|  a  cord  ? 
266  What  is  the  value  of  V-  of  3^-  X  f  X  §  of  10  ? 

267.  What  is  the  value  of  f  of  f  of  ^y  multiplied  by  § 
of  18  ? 

268.  What  is  the  product  of  ^  of  8^^  multiplied  by  f 
of  9i? 

269.  What  will  5^^  tons  of  hay  cost  at  $21^^  a  ton  ? 

270.  If  a  man  walks  4-i^^  miles  in  an  hour,  how  many  miles 
can  he  walk  in  }  of  S-j^^  hours  ? 


COMMON   FRACTIONS. 


81 


DIVISION. 

A  Fraction  divided  by  an  Integer. 

ORAL    EXERCISES. 

271.  What  is  §  divided  by  2  ? 

Solution.  —  J  divided  by 
2  is  f-^^,  equal  to  ^.,  Or,  as 
f  divided  by  2  is  the  same  as 
^  of  |.  we  have  f^g?  equal  to 

It  will  be  observed  by  the 
illustration  that  dividing  the 
numerator  by  2  halves  the 
number  of   fractional   units, 
while  their  size  remains  un- 
changed ;  and  that  multiplying  the  denominator  by  2  halves  the  size 
of  the  fractional  units,  while  the  number  remains  unchanged.    Either 
process  divides  the  fraction  by  2. 

272.  What  is  ^^  divided  by  3  ?     \ihj5? 

273.  Divide  |i  by  7  ;  Jf  by  9  ;    f  |  by  7  ;    ^^^^  by  11. 

274.  Divide  the  fractions  in  Exercises  99  to  103  by  2 ;  by 
3 ;   by  5 ;  by  6  ;    by  8  ;  and  simplify'-  the  result. 

275.  If  8  men  can  mow  |  of  a  field  in  a  day,  what  part  of 
it  can  one  man  mow  ? 

276.  How  is  -^(j  divided  by  5  ?     f  by  8  ?     f  by  9  ? 

277.  Divide  J  by  6;    H  by  7;    y^^  by  10;    u'hy  6. 

278.  If  7  pounds  of  coffee   cost  $2y'^,   how  much  will  1 
pound  cost  ? 

Solution.  —  $  2-^Q  =  $  ^1 ;   if  7   pounds  cost  $  f|,  1  pound  must 
cost|of$fi  orf^V 

279.  How  much  is  2tV  divided  by  7  ?     11^  divided  by  17  ? 
lOf  divided  by  25  ? 

280.  If  9  bushels  of  wheat  cost  $  19|,  what  is  the  cost  of 
1  bushel  ? 

a 


82 


COMMON   FRACTIONS. 


132.    Principle. 

A  fraction  is  divided  either  hy  dividing  its  numerator  or 
hj  7Rultiplying  its  denominator. 

WRITTEN    EXERCISES. 

281.    Divide  ^f  by  9. 


18 

23 


Q         18   -^ 
9  =  23 


2 


Or, 


1  8     .    0—11         _JL8_: 
2'3  ~  ^—  23X9  —  2  OT 


Solution.  —  As  a  fraction  is  di- 
vided by  dividing  its  numerator^ 
we  divide  the  numerator  by  9,  and 
have  2^3-.  Or,  as  a  fraction  is  divided 
2  by  multiplying  the  denominator, 
2^  3"  we  multiply  the  denominator  by  9, 
and  have,  as  before,  ■^^, 


282. 

Divide 

tV  ^y  12. 

283. 

Divide 

hi  ^y  49. 

284. 

Divide 

^^ybyll. 

285. 

Divide 

m  ^y  21. 

290. 

Divide  17f  by  6. 

6)17f 

943 
^48 

Divide  f  J  by  60. 
Divide  f  4.  by  16. 


286. 
287. 
288.    Divide 


Or 


13  J.. 


17 


—  XI 


-r- 


a  —   13-^  —13  9 


>43 


T^T  ^y  5- 

289.    Divide  f  f  J  by  75. 


Solution.  —  6  in  17f ,  2  times, 
and  5|  remainder.  5|  =  ^/, 
and  ^  divided  by  6  gives  ||, 
which,  united  with  the  partial 
quotient  2,  gives  2|-|. 

Or,  17|  zz:  J^p,  and  ^3.  di- 
vided by  6  gives,  as  before, 

m 


291. 


When  $  28  is  paid  for  2-i^  tons  of  hay,  what  part  of  a 
ton  will  $  1  purchase  ? 

292.  When  $  107^^^  is  shared  equally  by  7  men,  how  many 
dollars  will  each  man  receive  ? 

293.  The  product  of  two  numbers  is  30|^,  and  one  of  the 
numbers  is  9.     What  is  the  other  number  ? 

294.  When  12  cords  of  wood  cost  $  76^,  what  is  the  cost  a 
cord  ? 


COMMON  FRACTIONS.  83 

An  Integer  divided  by  a  Fraction. 
ORAL    EXERCISES. 

295.  How  many  fourths  of  a  dollar  in  $  1  ?     In  $  3  ? 

296.  How  many  times  is  J  in  1  ?     In  2  ?     In  3  ? 

297.  In  4  how  many  times  i?     J?     §?     J?     f? 

298.  At  $  J  a  pound,  how  many  pounds  of  tea  can  be  bought 
for  I  6  ? 

Solution.  —  As  many  pounds  as  times  f  in  6.     In  6  there  are  ^^, 
and  I  are  contained  in  ^-^  8  times.     Ans.  8  pounds. 

299.  In  how  many  days  can  a  horse  consume  4  bushels  of 
oats,  if  he  consume  ^^3^  of  a  bushel  a  day  ? 

300.  Divide  any  given  integer  by  the  fractions  in  Exercises 
27  to  33. 

301.  At   $1J  a  yard,   how  many  yards  of  cloth  can  be 
bought  for  $  10  ? 

302.  In  how  many  weeks  will  a  boy  earn  $  8,  if  he  is  paid 
$2|aweek? 

WRITTEN    EXERCISES. 

303.  Divide  65  by  f 

Q^  _  A  5  A  .      JL  5  6.  _^  5  _  g-j^  Solution.  —  65  r=  ^^ ;  and 

"I  is  contained  in  ^^  as  many 

^  1  3  times  as  5  in  455,  or  91  times. 

65  -^  f  =  ^^"^^  =  91  Or,  I  is  contained  in  65,  7 

times  65  times,  and  f ,  ^  of 
7  times  65  times,  or  91  times.  This  process  is  the  same  as  multiply- 
ing the  dividend  by  the  divisor  inverted. 

304.  Divide  49  by  Jf .  309.  Divide  98  by  H- 

305.  Divide  27  by  |J.  310.  Divide  128  by  S^\. 

306.  Divide  84  by  f  |.  311.  Divide  432  by  2§. 

307.  Divide  49  by  +f .  312.  Divide  100  by  If 

308.  Divide  60  by  j\.  313.  Divide  118  by  5|. 

314.    How  many  yards  of  cloth  at  $  |  a  yard  can  be  bought 
for  $56?  ,.^ 


84 


COMMON   FRACTIONS. 


315.  If   a  family  consumes  2|  pounds  of  sugar  a  week,  in 
what  time  will  it  consume  44  pounds  ? 

316.  There  is  a  board  17  feet  long,  which  I  wish  to  saw  into 
pieces  2f  feet  long.     What  will  be  the  number  of  pieces  ? 

A  Fraction  divided  by  a  Fraction. 
ORAL    EXERCISES. 

317.  How  many  times  f  of  a  yard  in  f  of  a  yard  ? 

318.  At  I  of  a  dollar  a  pound,  how  many  pounds  of  coffee 
can  be  bought  for  |  of  a  dollar  ? 

319.  How  many  times  f  in  |  ?     |  in  |  ?     f  in  Y-  ? 

320.  How  many  pounds  of  raisins  at  -^^  of  a  dollar  a  pound 
can  be  bought  for  f  of  a  dollar  ? 

Solution.  —  As  many  pounds  as  times  y%  in  f .     In  f  there  are  ^|-, 
and  y\  are  contained  in  ^|,  4  times.     Ans.  4. 

321.  How  many  times  ^V  in  f  ?     i  in  f  ?     |  in  ^3  ? 

322.  If  it  takes  f  of  a  yard  of  cloth  for  a  vest,  how  many 
vests  can  be  made  from  2J  yards  ? 


323. 

324. 

325. 

A- 

-i  =  ? 

i  - 

-|  =  ? 

1   - 

-4  =  ? 

i   - 

-f  =  ? 

«  - 

-i  =  ? 

2J- 

-t  =  ? 

H- 

-f  =  ? 

fT7- 

-f  =  ? 

2t- 

-f  =  ? 

^  - 

-§  =  ? 

5    - 

-§  =  ? 

H- 

-§  =  ? 

f  - 

-i  =  ? 

A- 

-|  =  ? 

H- 

-i  =  ? 

WRITTEN    EXERCISES 

326.    Di 

vide  1  by  §. 

Solution.  —  t  -^  I 

I  -  f  =  if  -  i^  =  if  =  I  =  li       •«  equivalent  to  ^  di- 
vided by  ^f  or  to  {^, 
Or,  ^  equal  to  |,  or  1^. 

1        5        ?A5       t  r  ^^1  is  III,  equal  to 

i,  or  H- 
Thia  is  the  same  as  multiplying  tlic  dividend   by  the  divisoi 
inverted. 


COMMON   FRACTIONS.  85 

327.  What  is  the  quotient  of  J  divided  by  f  ? 

328.  What  is  the  quotient  of  Jf  divided  by  /^  ? 

133.    Rule  for  Division  of  Fractions. 

Change  the  fractions,  if  necessary,  to  fractions  having  a 
common  denominator,  and  divide  the  numerator  of  the  divi- 
dend by  the  numerator  of  the  divisor.     Or, 

Invert  the  divisor,  and  proceed  as  in  multiplication  of  frac 
tions. 

Note  1.  —The  rule  is  general,  since  a  mixed  number  may  be  changed  to  an 
improper  fraction,  and  an  integer  may  be  expressed  in  a  fractional  form. 
Note  2.  —  A  fraction  having  a  fraction  in  one  or  both  of  its  terms,  as  in 

'    3  '  7    ^^  called  a  Complex  Fraction^  and  may  be  considered  an  expression 
3      £      s 
of  division. 

329.  Divide  f  f  by  \%,  334.    Divide  f  J  by  f  f . 

330.  Divide  if  by  f .  335.   Divide  9|  by  4J. 

331.  Divide  \^  by  If.  336.   Divide  f §|  by  iff 

332.  Divide  iJ  by  f.  337.    Divide  17  2^  by  5f. 

333.  Divide  ^j%  by  iff.  *    338.   Divide  lOOg  by  8f. 

339.  How  much  is  tea  a  pound  when  \^  of  a  pound  cost 
if  of  a  dollar  ? 

340.  How  much  is  hay  a  ton  when  |  of  a  ton  cost  $  llj  ? 

341.  If  a  man  walks  3f  miles  in  an  hour,  in  what  time  will 
he  walk  15^^^  miles  ? 

342.  Simplify  ||. 

Solution.  —  Consid' 

3 1  _  JL5_  -^-  1  =  IJl  x^  —  ^4  =  14-1-      erinff  the  fraction  as  an  ^ 
^|—    4*3  4'^8  —  32         -*-32  ^     .  o    ^'    -  - 

expression  oi  division, 

^^j  we  reduce  its  terms  to 

2"l^^'32"~l3"2"  '  improper  fractions,  and, 

dividing,  have  f|,  or  1  Jf . 

Or,  multiplying  both  terms  of  the  complex  fraction  by  12,  the  least 

common  multiple  of  the  denominators  of  the  fractional  parts,  we  have, 

as  before,  f|,  or  Ijf . 


86  COMMON   FRACTIONS. 

343.  Simplify  S.        345.    Simplify-^.        347.   Simplify—-. 

5  t  ^A 

12  5^  6# 

344.  Simplify—.        346.   Simplify  j|.       348.    Simplify -|. 

104 

349.  Eeduce  ^^  to  its  simplest  form. 

350.  If  a  man  can  do  J f  of  a  piece  of  work  in  23 f  days,  in 
what  time  can  he  do  the  whole  ? 

351.  At  the  rate  of  31  x\  miles  an  hour,  in  what  time  will  a 
train  of  cars  move  125^  miles  ? 


To  find  the  Fraction  one  Number  is  of  another. 

352.  One  is  what  part  of  2  ?     Of  3  ?     Of  4  ?     Of  5? 

353.  What  part  of  5  dollars  is  1  dollar  ?  2  dollars  ?  3 
dollars  ?     4  dollars  ? 

354.  What  part  of  7  miles  is  3  miles  ?  4  miles  ?  5  miles  ? 
6  miles  ? 

355.  What  part  of  5  is  §  ? 

Solution.  —  1  is  ^  of  5,  and  f  of  1  is  |  of  ^  of  5,  or  -^  of  5. 

356.  What  part  of  4  dollars  is  f  of  a  dollar  ? 

357.  What  part  of  ^^  is  ^\  ?     What  part  of  f  is  |,  or  |  ? 

358.  What  part  of  8  dollars  is  2J  dollars  ? 

134.  The  Fraction  one  number  is  of  another  is  found  bj 
dividing  the  number  denoting  the  jpart  ly  the  number  denot- 
ing the  whole. 

WRITTEN    EXERCISES. 

359.  What  part  of  125  is* 75  ?  363.   What  part  of  20  is  3]^? 

360.  What  part  of  7  is  G  ?  364.    What  part  of  90  is  6J? 

361.  What  part  of  12  is  |f  ?  365.'  J  J  is  what  part  of  |§  ? 

362.  What  part  of  186  is  93?  366.   ^J  is  what  part  of  J  J  ?. 


COMMON   FRACTIONS.  87 

367.  What  fraction  of  84  is  91  ? 

368.  What  fraction  of  3f  is  2^  ? 

369.  A  tree,  whose  height  was  85  feet,  was  broken  off  in  a 
gale,  55  feet  from  the  top.  What  part  of  the  tree  was  left 
standing  ? 

370.  A  sportsman  started  up  a  flock  of  144  birds,  and  shot 
§  of  them.     What  number  escaped  ? 

371.  If  25  yards  of  carpeting  cost  $  37.50,  what  should  35 
yards  cost  at  the  same  rate  ? 

372.  John  has  156  cents,  and  Peter  106  cents.  How  does 
Peter's  money  compare  with  John's  ? 

373.  If  a  man  can  do  a  piece  of  work  in  7f  days,  what  part 
of  it  can  he  do  in  6  J  days  ? 

374.  What  will  be  the  cost  of  ^  of  a  cord  of  wood,  when  12 
cords  cost  $68^? 

To  find  the  Whole  when  a  Fractional  Part  is  given. 

375.  4  is  ^  of  what  number  ?  6  is  ^  of  what  number  ?  8 
is  J  of  what  number  ?     3  is  ^  of  what  number  ? 

376.  I  spent  $  6,  which  was  ^  of  all  my  money  ;  how  much 
had  I? 

377.  I  lost  f  of  my  sheep,  but  had  8  left ;  how  large  a  flock 
had  I  at  first  ? 

378.  18  years  is  f  of  my  age  ;  how  old  am  I  ? 

Solution.  —  Since  18  years  is  3  fourths  of  my  age,  1  fourth  of  my 
age  is  J  of  18  years,  or  6  years;  4  fourths  of  my  age  is  4  times  6  years, 
or  24  years.     Ans.  24  years. 

379.  -«-  of  what  number  is  24  ? 

380.  32  is  f  of  what  number  ?     48  is  f  of  what  number  ? 

381.  36  is  ^^  of  what  number  ?    54  is  j%  of  what  number  ? 

382.  42  is  I  of  what  number  ?     84  is  /^  of  what  number  ? 

383.  63  is  ^j  of  what  number  ?    75  is  f -f  of  what  number  ? 

384.  How  far  is  it  to  Boston  if  f  the  distance  is  21  miles  ? 


88  COMMON   FRACTIONS. 

385.  My  house  is  insured  for  f  its  value,  or  for  $  4000  •, 
what  is  it  worth  ? 

386.  The  cost  of  my  buggy  is  |  the  cost  of  my  horse.     The 
horse  cost  $  360  ;  what  did  the  buggy  cost  ? 


WRITTEN    EXERCISES. 

387.  1728  is  If  of  what  number  ? 

1  Q  g  Solution.  —  Since  1728  is  if  of  an 

—iA^  X  19  =  2052        unknown  number,  ^^  of  the  unknown 
number  is  ^  of  1728,  or  108;  ||  of  the 
unknown  number  are  19  times  108,  or  2052.     Aus.  2052. 

388.  351  is  2^5  of  whaf  number  ? 

135.  To  find  the  whole  when  a  fractional  part  is  given, 
divide  the  part  hy  the  numerator  of  the  fraction^  and  multi- 
ply the  quotient  by  the  denominator, 

389.  1368  is  ^T  of  what  number  ? 

390.  891  is  4 J  of  what  number  ? 

391.  8000  is  \U  of  what  number  ? 

392.  What  is  f  of  a  number  if  273  is  f  J  of  it  ? 

393.  My  income  in  1881  was  $  1600,  or  ^  of  my  income 
in  1880  ;  what  was  the  falling  off  ? 

394.  jf  of  340  is  T^T  of  what  number  ? 

395.  f  is  ^^  of  what  number  ? 
/T$96.  7 J  is  tV  of  what  number  ? 

397.  Sold  goods  for  |462,  and  thereby  lost  ^  of  the  cost; 
what  was  the  cost  ? 

398.  A  man  failing  in  business  could  pay  me  only  $  3930, 
which  was  J  of  whftt  he  owed  me.     How  much  did  he  owe  me  ? 

399.  A  has  $  2000,  and  B  has  l-,^  times  as  much.  How 
much  has  B  ? 

400.  My  house  and  furniture  cost  %  10901,  and  the  furniture 
cost  f  as  much  as  the  house.     How  much  did  the  house  cost  ? 


COMMON   FRACTIONS.  89 

MISCELLANEOUS     EXERCISES. 

401.  At  $  f  a  yard^  how  many  yards  of  cloth  can  be  bought 
for$43i? 

402.  If  2^  pounds  of  butter  cost  $^f,  what  is  that  a 
pound  ? 

403.  How  many  yards  of  cloth  f  of  a  yard  wide  will  equal 
12  yards  f  of  a  yard  wide  ? 

404.  If  I  of  a  barrel  of  oatmeal  is  worth  $  5.60,  what  is  a 
barrel  worth  ? 

j/^405.    What  is  the  value  of  -|-  oi  12^  divided  by  i  of  8|  ? 

406.  What  number  must  be  multiplied  by  7|  that  the  pro- 
duct may  be  20  ? 

407.  A  can  walk  S^y  miles  in  60  minutes,  and  B  can  walk 
/t  as  fast  as  A.  How  long  will  it  take  B  to  walk  the  same 
distance  ? 

408.  If  $  106^  is  shared  equally  among  8  men,  how  much 
will  each  man's  share  be  ? 

409.  If  of  $  350  you  should  spend  $  125,  what  part  of  the 
money  would  remain  ? 

410.  There  is  a  board  19  feet  in  length,  which  I  wish  to 
saw  into  pieces  2f  feet  long.  What  will  be  the  number  af 
pieces,  and  how  many  feet  will  remain  ? 

411.  How  much  is  coal  a  ton  when  10^  tons  can  be  bought 
for  $  67^  ? 

412.  What  is  the  relative  value  of  ||  and  {^  ? 

>^413.    Eeduce  -^3- — ^— —  to  its  simplest  form. 

414.    What  number  must  2^  be  multiplied  by  to  give  4J  ? 
'    415.   Find  the  value  of  ^  of  Ij  X  4^  divided  by  /^  of  IJ 
X3i. 

416.  Jones  and  Smith  plowed  for  a  certain  sum  of  money  ; 
Jones  plowed  6 J  acres,  and  Smith  9f  acres.  What  should  be 
Smith's  money,  if  Jones's  share  is  $  20  ? 


90  COMMON   FRACTIONS. 

417.  What  number  must  be  taken  from  12|,  and  the  re- 
mainder multiplied  by  10|,  that  the  product  shall  be  50  ? 

418.  When  $236  are  paid  for  llf  acres,  what  will  20/^ 
acres  cost  ? 

419.  In  counting  my  money  I  found  I  had  $969,  which 
was  just  2 1  times  as  much  as  my  brother  had.  How 
many  dollars  had  I  more  than  my  brother  ? 


QUESTIONS. 

99.  What  is  a  fraction  ?  100.  What  ia  the  denominator  of  a  frac- 
tion ?  101.  What  is  the  numerator  ]  102.  What  are  the  terms  of 
a  fraction  ? 

105.  What  is  a  proper  fraction  1  106.  An  improper  fraction  ? 
107.   A  mixed  number  ] 

108.  How  may  a  fraction  be  regarded  1  109.  What  is  the  value 
of  a  fraction  ] 

110.  What  is  reduction  of  fractions?  112.  How  is  a  fraction 
reduced  to  larger  terms?  113.  When  is  a  fraction  reduced  to  its 
smallest  tenns  ?  114.  What  is  a  principle  of  fractions  ?  115.  How 
is  a  fraction  reduced  to  smallest  terms  ? 

116.  How  is  an  integer  or  mixed  number  reduced  to  an  improper 
fraction?  117.  How  is  an  improper  fraction  reduced  to  an  integer 
or  mixed  number  ? 

118.  When  have  fractions  a  common  denominator?  120.  What 
must  be  a  common  denominator  of  two  or  more  fractions?  121. 
How  are  fractions  reduced  to  their  least  common  denominator  ? 

122.  What  are  like  fractions?  123.  What  fractions  only  can  be 
added?  124.  How  do  you  add  fractions?  126.  What  fractions 
only  can  be  subtracted?  127.  How  do  you  subtract  one  fraction 
from  another  ? 

128.  How  is  a  fraction  multiplied  ?  129.  How  is  a  number  mul- 
tiplied by  a  fraction  ?  131.  How  is  one  fraction  multiplied  by  an- 
other? 

132.  How  is  a  fraction  divided?  133.  How  is  one  fraction  di* 
Vided  by  another  ? 


KEVIEW.  91 


REVIEW. 


ORAL   EXERCISES. 

136.    1.   Kame  the  only  even  prime  number. 

2.  Show  that  56  is  a  composite  number. 

3.  What  is  the  greatest  factor  common  to  16  and  24  ? 

4.  What  is  the  least  common  multiple  of  3,  4,  and  6  ? 

5.  Eeduce  9|  to  fifths  ;  7f  to  eighths ;  lO^^^  to  elevenths. 

6.  Eeduce  to  smallest  terms  |f ;  /^  ;  ^f ;  ^^^ ;  ^^. 

7.  A  has  worked  f  of  a  day,  and  B  i  of  a  day.     What  part 
of  a  day  have  both  worked  ? 

8.  I  of  a  pole  is  in  the  water,  |  in  the  mud,  and  the  rest  in 
the  air.     What  part  is  in  the  air  ? 

9.  James  is  18  years  old,  and  his  age  is  f  of  the  age  of  his 
father.     How  old  is  his  father  ? 

10.  Damon  sold  a  house  lot  for  $  30,  which  was  |  of  what 
it  cost  him.     What  was  the  cost  of  the  lot  ? 

11.  Multiply  J  by  6;  H  by  7;  5  by  f;  I  by  f. 

12.  If  a  man   can  mow  If  acres  of  grass  in  a   day,  how 
much  can  he  mow  in  6  days  ? 

13.  If  cloth  is  worth  -f^  of  a  dollar  a  yard,  what  is  |  of  a 
yard  worth  ? 

14.  Divide  ft  l^y  3  ;  ii  by  4;  8  by  §  ;  I  by  1 ;  2i  by  f. 

15.  Bought  5  bushels  of  wheat  for  $  7J.     What  was  it  a 
bushel  ? 

16.  How  much  is  J  of  8| ;  i  of  5f  ;  i  of  7f  ? 

17.  There  is  ^  pole  standing  so  that  f  of  it  is  in  the  water, 
^nd  f  as  much  in  the  mud.     How  much  is  in  the  mud  ? 

18.  If  f  of  a  bushel  of  grain  will  serve  a  horse  a  week,  how 
many  weeks  will  6|  bushels  serve  him  ? 

19.  If  1|  tons  of  hay  will  keep  a  cow  8  weeks,  how  manj 
cows  will  10  tons  keep  for  the  same  time  ? 

20.  What  number  is  that,  ^  of  which  exceeds  J  of  it  by  3  ? 

21.  The  difference  between  f  and  -/^  of  a  number  is   2. 
What  is  the  number  ? 


92  REVIEW. 

22.  If  to  William's  age  there  is  added  f  of  his  age  he  will  bt 
25  years  old.     What  is  his  age  ? 

23.  A  man  bought  a  coat  and  a  hat  for  $30.  The  hat 
cost  f  as  much  as  the  coat.     What  was  the  cost  of  each  ? 

24.  A  and  B  made  an  even  exchange  of  horses.  By  the 
trade  A  lost  $  40,  which  was  f  of  the  value  of  the  horse  he 
had  at  first.     What  was  the  value  of  each  horse  ? 

25.  Joseph  spent  /^  of  his  money  for  clothes.  He  then 
paid  away  $  6f  j  which  was  just  ^  of  all  he  had  left.  How 
many  dollars  had  he  at  first  ? 

26.  If  25  is  I  of  some  number,  what  is  |  of  the  same  num- 
ber ? 

27.  A  boy  spent  f  of  his  money,  and  then  had  given  him  I 
as  much  as  he  had  left.  What  part  of  his  money  did  he  then 
have  ? 

28.  What  cost  12  apples  at  the  rate  of  7  for  5J  cents  ? 

29.  If  6  men  can  do  a  piece  of  work  in  4f  days,  in  what 
time  will  1  man  do  it  ?     In  what  time  will  5  men  do  it  ? 

30.  If  you  can  travel  13  miles  in  3  hours,  how  many  miles 
can  you  travel  in  8  hours  ? 

31.  If  5  men  can  cut  6|  cords  of  wood  in  a  day,  how  many 
cords  can  7  men  cut  in  the  same  time  ? 

32.  If  A  can  do  a  piece  of  work  in  7  days,  and  B  in  5  days, 
what  part  of  it  can  both  do  in  a  day  by  working  together  ? 

33.  If  A  can  build  a  wall  in  3  days,  which  B  can  build  in  4 
days,  in  what  time  can  they  build  it  by  working  together  ? 

WRITTEN    EXERCISES. 

34.  Change  §j|^  to  smallest  terms. 

35.  Find  the  product  of  the  common  prime  factors  of  1728 
and  2880. 

36.  What  number  contains  each  of  the  prime  factors  of  243y 
972,  576,  but  no  others  ? 

37.  Show  that  J$f  is  greater  than  J  and  less  than  f . 

38.  Find  the  sum  of  Jg,  f,  and  ^7^. 


KEVIEW.  93 

39.  rind  the  sum  of  f ,  Ij,  V",  and  ISyV- 

40.  What  number  added  to  ^J^  will  make  16  ? 

41.  If  9J  yards  cut  from  a  roll  of  cloth  leaves  24j  yards, 
what  is  the  length  of  the  roll  ? 

42.  What  is  the  value  of  2f  times  i  oi  ^^? 

43.  Find  the  number  that  must  be  added  to  f  of  |  to  make 
I  off. 

44.  A  man  has  229J  pounds  of  honey,  which  he  wishes  to 
pack  in  boxes  containing  S^  pounds  each.  How  many  boxes 
will  he  require  ? 

45.  What  fraction  must  %^  be  divided  by  to  make  the  quo- 
tient 31^  ? 

46.  Find  the  least  number  which,  divided  by  6,  by  8,  and 
by  9,  gives  in  every  case  a  remainder  4. 

47.  The  tide  rose  f  of  a  foot  in  one  hour,  *||  the  next,  and 
f  the  third  hour.     How  much  did  it  rise  in  the  3  hours  ? 

-^48.    Change  ^  to  an  equivalent  fraction,  having  91  for  its 
denominator. 

49.   How  many  cakes  would  be  required  for  a  school  of  53 
children,  of  whom  27  are  boys,  if  each  girl  has  ^  of  a  cake,  and 
each  boy  ^  as  much  again  as  each  girl  ? 
^     "So.   How  many  times  does  t'^  +  i  +  i  contain  J  +  i  +  i  ? 

51..  What  is  the  product  of  §  of  V"  multiplied  1^7  j|  ? 

52.  What  fraction  of  3  bushels  is  -^^  of  2f  bushels  ? 

53.  The  difference  between  f  and  |  of  a  number  is  10. 
What  is  the  number  ? 

54.  Eeduce  f  of  ^^  of  ^  of  8|  X  V  ^^  ^  simple  fraction  in 
its  smallest  terms. 

55.  Sold  a  horse  for  $  105|,  which  was  |  of  its  cost.  What 
was  its  cost  ? 

56.  What  cost  13^^^  tons  of  coal  at  $  7|-  a  ton  ? 

57.  Bought  IJ  bushels  of  corn  for  $  3f f ;  what  was  it  a 
bushel  ? 


94  KEVIEW. 

58.  How  many  acres  of  land  at  1 17|  an  acre  can  be  bought 

59.  When  $  2.13  are  paid  for  |  of  a  cord  of  wood,  what  cost 
1  cord  ?     What  cost  lOj  cords  ? 

60.  A  boy,  after  spending  |  of  his  money  in  oranges,  finds 
that  ^  of  what  he  has  left  is  24  cents.  How  much  money  had 
he  at  first  ? 

61.  If  28  men  can  build  an  embankment  in  42  days,  in  how 
many  dsijs  can  f  as  many  men  build  it  ? 

62.  How  much  cambric  that  is  f  of  a  yard  wide  will  line 
6|  yards  of  cloth  that  is  1^  yards  wide  ? 

2-^ 

63.  What  is  the  value  of  (f  of  lj%)  -^  ^  ? 

64.  A  owns  j^^  of  a  store,  B  ^7-  and  C  the  remainder.  The 
profits  were  $  960.     What  was  the  share  of  each  ? 

65.  When  $  1 J  will  buy  |  of  a  gallon  of  oil,  what  part  of  a 
gallon  will  |  ^j  buy  ? 

66.  If  §  of  a  store  is  worth  $  3300,  what  is  the  value  of  j^^ 
of  the  same  ? 

67.  How  many  pounds  of  tea  can  be  bought  for  $  23f ,  if  7^ 
pounds  cost  $  5/^  ? 

68.  If  4  bushels  of  corn  cost  $  3^,  how  much  will  7  bushels 
cost? 

69.  A  will  do  §  as  much  as  B.  The  board  of  each  is  worth 
$  f  a  day.  If  B  is  paid  $1 J  a  day  and  board,  what  should  be 
paid  to  A  in  addition  to  his  board  ? 

70.  A  saves  I  of  his  income,  and  B,  having  the  same  in- 
come, spends  1 J  times  as  much  as  A,  and  finds  himself  $  62^ 
in  debt  at  the  end  of  the  year.  What  was  the  income  of 
each  ? 

71.  A  cistern  of  960  gallons  is  emptied  by  two  pipes,  A  and 
B,  in  5  and  7  minutes,  respectively.  How  much  water  will 
pass  through  each  if  both  are  opened  together  ? 

1/72.   What  is  the  value  of  ^    ^.    +  J  of  1§  +  (1^  -^  l^) 


DECIMAL   FRACTIONS. 


95 


DECIMAL   FRACTIONS. 

137.   A  Decimal  Fraction  is  a  fraction  whose  unit  is  di- 
vided into  tenths,  hundredths,  thousandths,  etc. 
.    138.   A  Decimal  is  a  decimal  fraction  expressed  without 
its  denominator  by  means  of  the  decimal  point  (Art.  20)c 

Thus, 

0.7,  0.05,  and  0.168  are  decimals. 

139.  A  Mixed  Decimal   is   an   integer  and  a  decimal 

Thus, 

17.06  and  5.305  are  mixed  decimals. 

140.  The  method  of  writing  decimals,  in  continuation 
of  the  notation  of  integers  (Art.  25),  is  shown  in  the  fol- 
lowing 


TABLE. 


Integer. 


Decimal. 


Place-names. 


Figures. 


■s  § 

0^    o    §e 

S     rf     o 


1 


^   F5 


13      r^         ^ 


"^    -r^      15 


7  9   3,154.63 


H 

,4 


Group-names.       Thousands. 


Thousandths.        Millionths.       Billionths. 


8 


The  number  in  the  table  is,  seven  hundred  ninety-three 
thousand  one  hundred  fifty-four,  and  six  hundred  thirty- 
eight  million  four  hundred  seventy-eight  thousand  one 
hundred  thirty-nine  billionths. 

Note  1.  —  The  place-names  above  and  below  ones  correspond  to  each  other, 
except  that  the  decimals  liave  the  termination  ths. 

Note  2.  —  A  decimal  with  a  common  fraction  is  called  a  complex  deciinal ; 
as,  0.16^,  reatl  sixteen  and  four-lifths  Imndredths. 


96  DECIMAL  FRACTIONS. 

,  141.   The  Denominator  of  a  decimal  is  1,  with  as  many 
ciphers  annexed  as  there  are  places  in  the  decimal.    Thus, 


\j.\j  xo  Y0>   v/.x«j  xo  YO^O'  ^'^-^-^   ■'^'^ 

1000^  ^^^ 

u.  v.vuy  -Lia  Yo 

WRITTEN     EXERCISES. 

Eead  the  following : 

1.   0.35                   8.   180.06i 

15. 

66766.71 

2.   0.035                  9.   4.7307f 

16. 

6676.671 

3.   0.7077              10.   473.7f 

17. 

14.000014 

4.   0.0095              11.  0.47378 

18. 

1.0000001 

5.   0.0007              12.   0.000931 

19. 

0.0000077 

6.   0.4003              13.   667.6671 

20. 

7000000.7 

7.   5.55555            14.   6.676671 

21. 

96.00300315 

Write  decimally : 

22.  Ten,  and  eleven  hundredths. 

23.  Sixty-seven,  and  seven  tenths. 

24.  Six  hundred  seven,  and  nine  hundred  six  thousandths. 

25.  One  thousand  five  hundred  forty-one,  and  one  hundred 
seventy -eight  ten-thousandths. 

26.  Sixteen,    and  one   hundred   twenty-two  hundred-thou- 
sandths. 

27.  Thirty-seven  thousand  four  hundred  eighty-eight  hun- 
dred-thousandths. 

28.  Nine  hundred  fifty-eight  thousand  four  hundred  thirty- 
three  millionth  s. 

29.  Four  thousand  four  hundred  four,  and  fifty-nine  and 
three-fourths  hundredths. 

30.  Sixty-eight,  and  three  hundred  three    thousand   eight 
hundred  seven  and  two-sevenths  miUionths. 

31.  Forty   million    three    hundred   three,    and   forty-three 
thousandths. 

32.  Forty-seven,  and  nine  million  nine  hundred  ninety-nine 
thousand  nine  hundj^.d  ninety-nine  ten-millionths. 


DECIMAL  FRACTIONS.  97 

Express  in  decimal  form  : 


33. 

m- 

38. 

7aV 

43. 

looo^ig-b. 

34. 

TOIJ-G- 

39. 

Th%Uh' 

44. 

^'^^t^^Atjd- 

35. 

AVAV 

40. 

9060501f|. 

45. 

6958Tig§^. 

36. 

^^TxfoW 

41. 

TOO%%(J^- 

46. 

T^XroTJTTTT- 

37. 

945j15Vtt- 

42. 

4167^^. 

47. 

T(J(TSuJ^T7ir' 

REDUCTION. 

A  Decimal  to  Smaller  or  Larger  Denominator. 

ORAL   EXERCISES. 

48.  How  many  tenths  in  1  ?  How  many  hundredths  ? 
How  many  thousandths  ? 

49.  How  many  hundredths  in  2  tenths  ?  In  3  tenths  ?  In 
9  tenths  ? 

50.  In  0.01  how  many  thousandths  ?  In  0.1  how  many 
hundredths  ?     How  many  thousandths  ? 

51.  How  many  tenths  in  0.50  ?  In  0.70  ?  In  0.900  ?  In 
2.50? 

52.  How  many  hundredths  in  0.600  ?    In  0.850  ?   In  6.500  ? 

142.    Principle. 

Annexing  a  cipher  to  a  decimal,  or  removing  a  cipher  from 
the  right  of  a  decimal,  does  not  change  its  value*  (See  Arts. 
Ill,  114.) 

WRITTEN    EXERCISES. 

53.  Write  0.16,  3,  and  1.014  as  decimals  having  the  least 
common  denominator. 

0.16  =  0.160  Solution.  — "YhQ   smallest   order  of  deci- 

3.       =  3.000  mals  in  the  given  numbers  is  thousandths. 

1.014  0. 16  expressed  as  thousandths  is  0.1 60,  and 

3.  expressed  as  thousandths  is  3.000. 
the  requi'^d  decimals  are  0.160,  3.000,  and  1.014. 

7 


98  DECIMAL   FKACTIONS. 

54.  Eeduce  71.500  to  tenths. 

55.  Change  19,  43.6,  0.64,  and  53  to  thousandths. 

56.  Express  15.600,  4.7,  and  13  as  hundredths. 

57.-  Eeduce  18.0156,  401.6,  and  176.4700  to  decimals  having 
the  least  common  denominator. 

A  Decimal  to  a  Common  Fraction. 

58.  How  many  halves  in  .5  ?     In  .50  ? 

59.  How  many  fourths  in  .25  ?     In  .75  ?     In  .750  ? 

60.  How  many  fifths  in  0.4  ?     In  0.6  ?     In  0.8  ? 

61.  How  many  twentieths  in  0.05  ?     In  0.15  ?     In  0.45  ? 

WRITTEN    EXERCISES. 

62.  Change  .355  to  a  common  fraction  in  its  smallest  terms. 

oF,K  —  _3_5_5_  _7_1_  Solution. — As  .355    is   355    tbou- 

»ooo  —    10  0  0    —   2  0  0 

sandths,  it  may  be  written  -f^-Q,  which, 

changed  to  its  smallest  terms,  is  -jy^-. 

63.  Change  .225  to  a  common  fraction  in  its  smallest  terms. 

64.  Change  .875  to  a  common  fraction  in  its  smallest  termg. 

143.    To  change  a  decimal  to  a  common  fraction  : 

Rule. 

Omit  the  decimal  point,  write  the  denominator,  and  change 
the  fraction  to  its  smallest  terms, 

Eeduce  to  common  fractions  in  smallest  terms : 

65.  .025  70.   .375  75.   9.37i 

66.  .561  71.   .368  76.   115.875 
'      67.    .054                  72.    11.75  77.   .01375 

68.  M^  73.   4.43|  78.   200.96 

69.  .625  74.    .09375  79.   .015625 

A  Common  Fraction  to  a  Decimal. 

80.  How  many  tenths  in  1  ?     In  J  ?     In  ^  ?     In  f  ? 

81.  How  many  hundredths  in  1  ?     In  J  ?     In  f  ?     In  g  ? 

82.  How  many  thousandths  in  1  ?     In  J  ?     In  J  ?     In  |  ? 


DECIMAL   FRACTIONS.  99 

WRITTEN    EXERCISES. 

83.  Change  f  to  a  decimal. 

8)  5.000  Solution.  —  ^  is  equal  to|  of  5.     5  equab  60 

.625  tenths,  or  5.0  ;  -^  of  50  tenths  is  6  tenths,  with 

2  tenths,  equal  to  20  hundredths,  remaining.  ^  of  20  hundredths  is 
2  hundredths,  with  4  hundredths,  equal  to  40  thousandths,  remain- 
ing.    ^  of  40  thousandths  is  5  thousandths.     Ans.  .625. 

84.  Change  f  to  a  decimal. 

85.  Change  ^y  to  a  decimal  of  four  places. 

11)  8.0000  Solution.— -f^  IS   equal   to  Jy  of  8.     As 

.7272t\  there  are  to  be  four  places  in  the  decimal, 

we  annex  four  decimal  places  of  ciphers  to 
the  numerator,  reducing  it  to  8.0000.     j\  of  8.0000  is  .7272-i\. 

86.  Change  ^^  to  a  complex  decimal  of  three  places. 

87.  Change  ;^  to  a  complex  decimal  of  four  places. 

144.    To  reduce  a  common  fraction  to  a  decimal : 
Rule. 

Annex  decimal  ciphers  to  the  numerator,  divide  hy  the  de- 
nominator, and  point  off  as  ma7iy  decimal  figures  in  the  quo- 
tient as  there  are  ciphers  annexed. 

Note  1.  — When  the  division  does  not  terminate,  or  has  been  carried  as  far 
as  is  desirable,  the  remainder  may  be  expressed  as  a  common  fraction  and  made 
a  part  of  the  result. 

Note  2.  —  When  an  approximate  result  is  sufficient,  a  fraction  of  ^  or  more 
than  ^  in  a  result  may  be  rejected,  and  the  last  fi^re  of  the  decimal  be  made  to 
express  1  more.     Thus, 

.7272  ft:  approximately  expressed  to  the  nearest  ten-thousandth  is  .72X3. 

Note  3.  —  When  the  fraction  as  a  part  of  a  decimal  is  unimportant,  it 
may  be  omitted,  and  the  incompleteness  of  the  result  simply  marked  by  +. 
Thus, 

.7272+  may  be  written  instead  of  .7272A. 

Eeduce  to  decimals : 


88.    iJ. 

91-    U- 

94.   4g^. 

89.    ^. 

92.    i^. 

95.   2i^. 

90.    i^. 

93.  m- 

96.    Sxis- 

100  DECIMAL   FRACTIONS. 

97.  Reduce  ^^  to  a  complex  decimal  of  four  places. 

98.  Reduce  j\  to  the  nearest  ten-thousandth. 

99.  Reduce  |f  to  an  approximate  decimal  of  four  places. 

100.  Reduce  -^-^  to  a  complex  decimal  of  four  places. 

101.  Reduce  /y  to  the  nearest  millionth. 

102.  Change  to  the  nearest  thousandth  f ,  §^,  and  ^f  ^y,  and 
find  the  sum  of  the  results. 

103.  Reduce  to  the  nearest  ten-thousandth,  and  add,  15f, 
20|,  12|,  and  2.68. 

104.  Reduce  3^^  to  the  nearest  millionth,  and  subtract  the 
result  from  15.057. 

145.  For  rules  for  Addition  and  Subtraction  of  Deci- 
mals, see  Arts.  38  and  47. 

MULTIPLICATION. 

ORAL    EXERCISES. 

105.  How  much  is  3  times  2  tenths  ?     3  times  0.3  ? 

106.  How  much  is  7  times  1  hundredth  ?  7  times  -^^-^  ? 
9  times  0.06  ? 

107.  How  much  is  ^^^  of  1  ?     Of  3  ?     -^^  of  ^Jq  ?     iV  of  .3  ? 

108.  t^^XtV?     AXtSu?   3x.3?     4X.3?    .04  x  .3  ? 

109.  How  many  places  of  decimals  in  the  product  when 
tenths  are  multiplied  by  tenths  ?  When  hundredths  are  mul- 
tiplied by  tenths  ? 

WRITTEN    EXERCISES. 

lia   Multiply  31.5  by  .07. 

31.5  Solution.  —  As  .07  is  the  same  as  ^^  of  7,  31.5 

^07  multiplied  by  .07  is  .the  same  as  y^  of  7  times  31.5. 

2  2Q5  7  times  31.5  ==  220.5,  and  y^  of  220.5,  which  is 

found  by  removing  the  decimal  point  two  places 

to  the  left  (Art.  75),  is  2.205. 

ill.   What  is  the  product  of  671  by  .305  ? 
112.   What  is  the  product  of  18.72  by  7.1  ? 


DECIMAL  FRACTIONS.  >^   ^  ],./.   IPl, 

146.    Rul^  for  Multiplication  of  Decimals. '  ' 

Multiply  as  integers  j  and  point  off  as  many  figures  for  deci- 
mals in  the  product  as  there  are  decimal  places  in  both  factors. 

Note.  —  If  there  are  ni)t  jBgures  enough  in  the  product,  supply  the  deficiency 
by  prefixing  ciphers. 

113.  114. 

Multiply         .126  3.18 

By                  ^  ■                 .00029 

756  2862 

126  636 


Product     .002016  .0009222 

115.  Multiply  5.64  by  45.        120.  Multiply  .563  by  47. 

116.  Multiply  96.5  by  100.      121.  Multiply  19634  by  .0073. 

117.  Multiply  6.34  by  .0023.  122.  Multiply  .0703  by  .0055. 

118.  Multiply  42.2  by  2.004.    123.  Multiply  .0505  by  .001. 

119.  Multiply  1671  by  .013.     124.  Multiply  .0076  by  .017. 

125.  What  is  the  product  of  one  million  by  one  millionth  ? 

126.  What  is  the  cost  of  35.75  yards  of  cloth,  %  4.50  a  yard  ? 

127.  How  much  must  be  paid  for  13.375  cords  of  wood,  at 
%  4.62  a  cord  ? 

128.  What  is  the  product  of  one  hundred  one  thousandths 
by  ten  thousand  one  hundred  one  hundred-thousandths  ? 

DIVISION. 

ORAL   EXERCISES. 

129.  How  many  times  2  tenths  in  8  tenths  ?     3  tenths  in 
9  tenths  ? 

130.  How  much  is  ^  of  8  tenths  ?     J  of  9  tenths  ? 

131.  How  much  is  \  of  15  hundredths  ?     i  of  .24  ?     J  of 
.45? 

132.  Divide  .8  by  .2 ;  .63  by  7  ;  .63  by  .07. 

133.  The  product  of  two  factors  is  .72,  and  one  of  the  fac- 
tors is  9.     What  is  the  other  factor  ? 


102  DECIMAL   FRACTIONS. 

134.  Tlie  product  of  two  factors  is  .72,  and  one  of  the  fac< 
tors  is  .9.     What  is  the  other  factor,? 

135.  In  .72  divided  by  9,  how  many  places  of  decimals  are 
there  in  the  quotient  ?     In  .72  divided  by  .9  ? 

WRITTEN    EXERCISES. 

136.  Divide  46.48  by  .4. 

\A.)  46f4.8  Solution.  —  Both  dividend  and  divisor  may  be 

116.2  multiplied  by  10  without  changing  the  quotient 

(Art.  72).  This  is  done  by  moving  the  point 
one  place  to  the  right  (Art.  75).  Dividing  as  in  integers,  and  placing 
the  quotient  point  under  the  new  dividend  point,  we  have  the  quo- 
tient 116.2. 

137.  Divide  .7935  by  .23. 

3.45      ' 

X^o.)  pv.oo  Solution.  —  To  make  the  divisor  an  integer, 

^"  we  multiply  both  divisor  and  dividend  by  100, 

103  by  moving  the  point  in  each  two  places  to  the 

92  right.     Dividing  as  in  integers,  and  placing  the 

quotient  point  over  the  new  dividend  point, 

we  have  3.45  as  the  quotient  required. 


115 
115 


138.  Divide  1.264  by  4. 

139.  Divide  .00115  by  .05. 

147.    Rule  for  Division  of  Decimals. 

If  the  divisor  is  an  integer,  divide  as  in  integers,  and  point 
off  as  many  decimal  figures  in  the  quotient  as  there  are  such 
places  in  the  dividend*     •*'  v 

If  the  divisor  is  a  decimal,  make  it  an  integer  by  moving 
the  decimal  point  a  sufficient  number  of  places  to  the  right ; 
move  the  decimal  point  in  the  dividend  as  many  places  to  the 
right,  annexing  ciphers,  if  necessary,  and  then  divide. 


/ 


DECIMAL   FRACTIONS.  103 

Note  1.  —  When  the  division  does  not  terminate,  or  has  been  carried  as  far 
as  is  desirable,  the  remainder  may  be  expressed  as  a  common  fraction  and  made 
a  part  of  the  result ;  or,  the  fraction  may  be  rejected  and  an  approximation  ex- 
pressed to  the  nearest  decimal ;  or,  the  sign  +  may  be  nsed  to  mark  the  incom- 
pleteness of  the  division  (Notes,  Art.  144). 

Note  2.  —  When  the  divisor  is  an  integer,  the  decimal  point  in  the  quotient 
must  invariably  be  placed  directly  under  or  over  that  of  the  dividend.  If  it  is 
desirable  to  divide  without  changing  decimal  points,  it  maybe  done,  care"  being 
talj:en  to  give  the  quotient  as  many  decimal  places  as  those  in  the  dividend  ex 
ceed  those  in  the  divisor. 

140.  Divide  37.4  by  4.5  to  141.  Divide  6.0512  by  3.7  to 
three  decimal  places.  the  nearest  thousandth. 

8.311+  1.635 

4i5.)  37^4.000  3i7.)  6^0.512 

360  37 

140  235 

135  222 


50  131 

45  111 


50  202 

45  185 

"6  17 

142.  Divide  783.5  by  6.25.      147.  Divide  .817  by  .9147.     -r 

143.  Divide  .0189  by  .025.      148.  Divide  1.365  by  1000. 

144.  Divide  .01001  by  .001.   149.  Divide  72  by  .018. 

145.  Divide  1.0665  by  .00135. 150.  Divide  555  by  .0037. 

146.  Divide  .08748  by  1.08.     151.  Divide  .0016016 by  .00143. 

^152.  Divide  ninety-five  and  three  tenths  by  two  hundred 
Bixty-four  thousandths,  to  the  nearest  thousandth. 

153.  Divide  one  hundred  eighT:  and  twenty-nine  thou- 
sandths by  seven  and  two  tenths,  to^'^hfee  decimal  places. 

154.  If  a  man  can  walk  three  and  seventy-five  hundredths 
miles  in  an  hour,  in  how  many  hours  can  he  walk  seven  hun* 
dred  eighty-seven  and  five-tenths  miles  ? 


104  DECIMAL  FRACTIONS. 

MISCELLANEOUS     EXERCISES. 

155.  Write  four  hundred  three  ten-thousandths ;  four  nuiv 
dred,  and  three  ten-thousandths. 

156.  Write  and  read  :  0.00567  ;  2.13007 ;  1.00157. 

157.  Express  as  decimals  :  y^V^Ty ;  t^o^tt  5  TTTxrWiy' 
'4.  158.   rind  the  mixed  decimal  equal  to  4|  +  ^J. 

159.  From  45  subtract  36.00073. 

160.  Find  the  least  fraction  which  is  to  be  added  to  the  sum 
of  25.7,  8.389,  and  23.056  to  make  the  result  an  integer. 

'    161.   Find  the  difference  between  6346  and  .6346  ;  4.2  and 
.0042  ;  .0000005  and  .00005. 

162.  Simplify,  and  express  the  result  in  a  decimal  form, 
31  +  17^  +  476  +  3.125. 

163.  Change  .03125  to  a  common  fraction  in  its  smallest 
terms. 

164.  Change  4,  2f ,  17,  .136,  and  .0408  to  equivalent  decimals 
having  a  common  denominator,  and  find  their  sum. 

f*     165.    If  the   year   is   considered    365.25  days,    instead    of 
365.242264,  how  great  will  be  the  error  in  1880,  years  ? 

166.  The  dividend  is  7423.973,  the  quotient  12.130,  and  the 
remainder  .413.     What  is  the  divisor  ? 

.-^^167.   Divide  $  7498.70  among  A,  B,  and  C,  so  that  A  shall 
have  just  $749.83  more  than  each  of  the  others. 

168.  What  is  the  value  of  20004  +  (20.104  X  5.07)  - 
(6.44  ~  .005)  ?       -  / ..  y  /  v^   9>7  ?  r 

169.  Add  2§,  4U,  and  51.652,  expressing  the  sum  to  the 
nearest  thousandth. 

170.  A  man  willed  his  property  to  his  three  sons,  —  to  the 
youngest  he  gave  $968.49;  to  the  second,  3.4  times  as  much 
as  to  the  youngest ;  and  to  the  eldest,  3.7  times  as  much  as  to 
the  second.     Kequired  the  value  of  his  property. 

171.  If  a  train  of  cars  moves  at  the  rate  of  30.25  miles  an 
hour,  find  to  the  nearest  ten-thousandth  how  many  hours  it 
will  take  to  go  150.75  miles. 


DECIMAL   FRACTIONS.  105 

(l72.   Beduce  to  its  simplest  form  -1  -  ,  ^  ^^  ^^,^  ^  ^^^.    j 

173.  Dividing  a  certain  sum  by  .027,  the  quotient  is  6116. 
and  the  remainder  .003.     What  is  the  dividend  ? 

174.  Bought  wheat  at  $  0.94  per  bushel,  to  the  amount  of 
$  59.22,  and  sold  it  for  |  70.56.    What  was  made  on  a  bushel  ? 

175.  What  is  the  cost  of  60.5  tons  of  coal  when  0.9  of  a  tor 
costs  $  6M  ? 

176.  Find  .3  of  .064f  of  $  1728. 

177    Divide  (12^8  -^  .16)  by  (.128  ~  .0016).    L 
f  178.   What  part  of  .84  is  .012  ? 
y    179.    If  .6875  of  a  gallon  of  wine  costs  $3.75,  what  will 
40.25  gallons  cost  ? 

180.  If  I  of  an  acre  costs  $  15|,  what  will  46.78  acres  cost  ? 

181.  If  one  pound  costs  $  0.18f ,  how  many  pounds  can  be 
bought  for  $  14.40  ? 

182.  If  35.84  cubic  feet  of  water  weigh  a  ton,  what  will  be 
the  weight  of  2458.6  cubic  feet  ? 

183.  If  48  is  .08  of  some  number,  what  is  .7  of  it  ?      • 

184.  The  product  of  three  factors  is  5.76 ;  one  of  them  is 
.024,  another  is  .06 ;  find  the  third. 

185.  How  many  times  is  the  difference  of  (f  -r-  0.33J)  and 
(0.87 j^  -^  J)  contained  in  their  product  ? 

QUESTIONS. 

137.    What  is  a  decimal  fraction?     138.    What  is  a  decimal  ?     139 
A  mixed  decimal  ?     141.    What  is  the  denominator  of  a  decimal  ? 

142.  What  is  the  prin«iple  in  relation  to  annexing  or  cutting  off  a 
cipher  from  the  right  of  a  decimal  ? 

143.  How  is  a  decimal  reduced  to  a  common  fraction  ?     144.    How 
Is  a  common  fraction  reduced  to  a  diecimal  ? 

146.    How  do  you  multiply  in  decimals  ?     147.   How  do  you  di- 
nde  in  decimals  1 


106 


UNITED   STATES   MONEY. 


UNITED    STATES    MONEY. 


10  mills  (m.)  are  1  cent,  c.  or  /. 
10  cents  "    1  dime,  d. 

10  dimes  "    1  dollar,  $. 

10  dollars  "    1  eagle,  e. 

149.  Coins  are  pieces  of  metal 
stamped  by  authority  of  the  govern- 
ment to  circulate  as  money. 

150.    The  principal  Coins  of  the  United  States  are :  — 
Bronze,  —  the  cent ;  Nickel,  —  the  five-cent ;  Silver, 

—  the  dime,  quarter-dollar,  half-dollar,  and  dollar;  Gold, 

—  the  dollar,  quarter-eagle,  three-dollar,  half-eagle,  eagle, 
and  double-eagle. 


UNITED    STATES   MONEY.  107 

151.  In  ordinary  business  transactions,  eagles,  dimes^ 
and  mills  are  rarely  mentioned,  eagles  being  expressed  as 
dollars,  dimes  as  cents,  and  mills  as  a  fraction  of  a  cent. 
Thus, 

3  eagles,  2  dollars,  5  dimes,  8  cents,  5  mills  are  written, 

$32.58^-. 

152.  To  change  cents  to  millSy  multiply  by  10. 
To  change  dollars  to  centSy  midtijply  by  100. 
To  change  dollars  to  nfiills,  multijply  by  1000. 

Thus, 

45  cents  =  450  mills ;  $  5  =  500  cents,  or  5000  mills. 

153.  To  change  mills  to  cents,  divide  by  10. 
To  change  cents  to  dollars,  divide  by  100. 
To  change  mills  to  dollars,  divide  by  1000. 

Thus, 

850  mills  =  $0.85;  365  cents  =  $  3.65  ;  47000  mms  =  $  47. 

154.  The  dollar  being  the  unit,  of  which  cents  are  hun- 
dredths and  mills  are  thousandths,  it  follows  that 

All  the  rules  for  the  processes  in  decimals  are  applicable 
to  processes  in  United  States  money, 

ORAL   EXERCISES. 

1.  How  many  mills  in  7  cents  ?     In  7  cents  4  'mills  ? 

2.  How  many  cents  in  70  mills  ?     In  75  mills  ? 

3.  Change  $  5  to  cents.         7.   Change  500  cents  to  dollars. 

4.  Change  $  5  to  mills.         a    Change  5000  mills  to  dollars. 

5.  Change  $3.50  to  cents.      9.    Change  350  cents  to  dollars. 

6.  Change  $  6.37  to  mills.    10.    Change  6370  mills  to  dollars. 
11.   In  changing  cents  to  dollars,  how  many  places  to  the 

left  is  the  decimal  point  moved  ? 


108  UNITED   STATES   MONEY. 

12.  In  changing  mills  to  dollars,  how  many  places  to  the 
left  is  the  decimal  point  moved  ? 

13.  Paid  for  a  bag  of  flour  $  1.25,  for  coffee  40  cents,  and 
for  soap  20  cents.     How  much  was  paid  for  the  whole  ? 

14.  A  boy  earned  one  week  $  3.50,  and  the  next  75  cents. 
How  much  did  he  earn  in  all  ? 

15.  Jane  has  60  cents,  Alice  15  cents,  and  Henry  $  1.20. 
How  much  have  they  all  ? 

16.  Bought  an  arithmetic  for  $  1.25,  and  gave  a  two-dollai 
bill  in  payment.     What  change  should  I  receive  ? 

17.  Bought  a  plow  for  $  12.50,  and  sold  it  for  $  15.  How 
much  did  I  make  ? 

18.  How  much  is  the  profit  on  flour  bought  for  $  7.50,  and 
6old  for  $  9.25  ? 

19.  If  a  man  can  earn  $  1.50  in  one  day,  how  much  can  he 
earn  in  3  days  ? 

20.  How  much  must  be  paid  for  5  pairs  of  shoes  at  $  2.25  a 
pair  ? 

21.  At  25  cents  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $  1.75  ? 

22.  At  50  cents  a  bushel,  how  many  bushels  of  apples  can 
be  bought  for  $  4.75  ? 

WRITTEN    EXERCISES. 

23.  Bought  a  coat  for  $13.50,  a  vest  for  $2.63,  a  hat  for 
$  5,  and  a  pair  of  boots  for  $  6.13.     What  was  the  amount  ? 

24.  A  man  paid  for  tea  63  cents,  for  butter  80  cents,  for 
flour  $  7.50,  and  for  other  articles  $  32.  What  did  he  pay  for 
the  whole  ? 

25.  Mr.  Avery  bought  a  farm  for  $6500,  and  paid  down 
$1356.85.     How  much  more  had  he  to  pay  ? 

26.  How  much  will  be  received  for  56  pounds  of  crackers 
at  14  cents  a  pound,  and  128  loaves  of  bread  at  9  cents  a  loaf  ? 

27.  How  many  yards  of  cloth  at  19  cents  a  yard  can  bo 
bought  for  $47.50? 


UNITED   STATES   MONEY. 


109 


28.  A  man  bought  491  bushels  of  corn  at  81  cents  a  bushel. 
He  used  29  bushels,  and  sold  the  rest  at  95  cents  a  bushel. 
How  much  did  he  make  ? 

29.  Bought  a  house  and  farm  of  36  acres  for  $  7975.  The 
house  is  worth  $  4560.  What  is  the  value  of  the  land  an  acre, 
k>  the  nearest  cent  ? 

30.  A  farmer  sold  187  acres  of  land  at  $  37.50,  and  131 
acres  at  $  63  an  acre.  At  what  rate  an  acre,  to  the  nearest 
cent,  did  he  sell  the  whole  ? 

31.  At  67  cents  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $  29.70  ? 

32.  At  9  cents  a  pound,  how  many  pounds  of  beef  can  be 
bought  for  $  68.40  ? 

33.  When  7  barrels  of  onions  are  worth  $  26.25,  what  are 
43.50  barrels  worth  ? 

34.  Bought  land  for  $1040.98,  and  sold  35.45  acres  at 
$  20.40  an  acre ;  the  rest  was  worth  to  me  $  20.50  an  acre. 
How  many  acres  did  I  buy  ? 


ALIQUOT    PARTS. 

155.  Aliquot  Parts  of  a  number  are  such  parts  of  the 
number  as  will  exactly  divide  it.     Thus, 

2,  2|-,  3J,  and  5  are  aliquot  parts  of  10. 

156.  Computations  are  often  abridged  by  use  of  the 
following 


Aliquot  Pai 

50    cents  =  -J  of  f  1. 

^ts  of  a  Dollar. 

12J  cents  =  |  of$l. 

33J  cents  =  l  of  $  1. 

10    cents  =  yVo^^  1- 

25    cents  =  1  of  $1. 

8 J  cents  =  ^i^  of  $  1. 

20    cents  =  ^  of  $  1. 

6^  cents  =  3^^  of  $1. 

16f  cents  =  1  of  $1. 

6    cents  =  tjV  0^  ^  1' 

110  UNITED    STATES   MONEY. 

ORAL    EXERCISES. 

35.  What  will  42  yards  of  cloth  cost  at  16  §  cents  a  yard  ? 

Solution.  —  16f  cents  is  -^  of  $  1.    Since  one  yard  costs  $  |,  42  yards 
will  cost  42  times  $  -J,  or  $  \%  or  $  7. 

36.  At  121-  cents  a  pound,  what  will  96  pounds  of  sugai 
cost? 

37.  At  25  cents  a  box,  how  much  will  60  boxes  of  straw 
berries  cost  ? 

38.  At  12^  cents  a  pound,  how  many  pounds  of  sugar  can 
be  bought  for  $  12  ? 

Solution.  —  At  12^  cents  a  pound,  $1  will  buy  8  pounds,  and  $12 
will  buy  12  times  8  pounds,  or  96  pounds. 

39.  At  25  cents  a  box,  how  many  boxes  of  strawberries  can 
be  bought  for  $15? 

40.  At  33  J-  cents  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $  9.33^  ? 

41.  At  50  cents  a  gallon,  how  many  gallons  of  molasses  can 
be  bought  for  1 11.50  ? 

42.  At  20  cents  a  dozen,  how  many  dozen   eggs   can  be 
bought  for  $  10.80  ? 

WRITTEN     EXERCISES. 

43.  What  will  1832  pairs  of  shoes  cost  at  $  1.75  a  pair  ? 

Solution.  —  1832   pairs   at  f  1 

will  cost  $1832  ;  at  $^,  will  cost 

J  of  $  1832,  or  $  916;  and  at  $  \, 

will   cost   J   of  $916,    or   $458, 

$3206=        "        $T75  $1832 +  $916 +  $458  =$3206. 

44.  What  will   144   bushels   of   wheat   cost   at   $  1.37 J  a 
bushel  ? 

45.  What  will  84  yards  of  cloth  cost  at  66J  cents  a  yard  ? 


1 1832  = 

cost  at  $  1 

916== 

a 

.50 

458  = 

cc 

.25 

UNITED    STATES   MONEY.  Ill 

46.  What  will  60  pairs  of  boots  cost  at  $  3.87^  a  pair  ? 

47.  At  $  5.12^  a  dozen;  how  many  dozen  pocket-knives  can 
be  bought  for  $502.25? 

48.  At  $1.62J  a  pair,  how  many  pairs  of  shoes  can  be 
bought  for  $29.25? 

49.  At  $  0.87^  each;  how  much  must  be  paid  for  56  caps  ? 

50.  At  $2.25  a  day,  how  much  will  a  man  earn  in  302 
days  ? 

51.  At  37^  cents  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $  793.87J  ? 

52.  What  will  575  njelons  cost  at  $  16  a  hundred  ? 

Solution.  —  575  is  equal  to  5.75,  or  5f  hundreds.     5|  hundreds,  at     y~' 
$  16  a  hundred,  will  cost  5f  times  •$  16,  or  $  92. 

53.  What  will  1360  herrings  cost  at  37^  cents  a  hundred  ? 

54.  What  will  650  melons  cost  at  $  12.50  a  hundred  ? 

55.  What  will  2250  feet  of  boards  cost  at  $  20.50  a  thou- 
sand  feet  ? 

Solution.  —  2250  is  equal  to  2.250,  or  2 J  thousands  ;  and  2^  times 
$20.50  =  $46.12^. 

56.  How  much  will  5650  feet  of  plank  cost  at  $  44  a  thou- 
sand feet  ? 

57.  What  is  the  cost  of  4565  feet  of  joist  at  $  23  a  thousand 
feet ;  13640  feet  of  boards  at  $  53.55  per  thousand  feet ;  and 
15250  shingles  at  $  4.50  per  thousand  ? 

5a  If  18500  bricks  are  sold  for  $  155.40,  what  is  the  cost 
per  thousand  ? 

Solution.  —  18500  =  18.5  thousands.  Since  18.5  thousands  cost 
$  155.40,  1  thousand  will  cost  $  155.40  -^  18.5,  or  $  8.40. 

59.  If  8375  feet  of  spruce  scantling  cost  $  lOO.SO^  what  is 
the  cost  of  a  thousand  feet  ? 


112 


UNITED    STATES   MONEY. 


60.  If  650  melons  cost  $  81.25;  what  is  the  cost  a  hundred  ? 

61.  Bought  a  farm,  containing  75.8  acres,  at  $  31.50  an 
acre,  and  sold  it  for  $  2274.     What  was  the  loss  per  acre  ? 

ACCOUNTS  AND  BILLS. 

157.  An  Account  is  a  record  of  articles  bought  or  sold, 
cash  paid  or  received,  or  services  rendered. 

158.  A  Debtor  is  one  who  owes  a  debt,  and  a  Creditor  is 

one  to  whom  a  debt  is  owed. 

159.  A  Bill  is  a  written  statement  of  an  account  made 
out  by  the  creditor  for  the  debtor. 

A  bill  is  Receipted  when  its  payment  is  acknowledged  in 
writing  by  the  creditor,  or  by  some  one  authorized  to  sign 
for  him. 

Note.  —  Dr.  is  for  "debtor,"  Cr.  for  "creditor,'  M.  for  "thousand," 
and  (aX  for  "  at." 


WRITTEN    EXERCISES. 

Copy  and  find  the  amounts  and  balances  of  the  following 
accounts  and  bills  :  — 

62. 

Itt  %amxd  foitfe   JAMES    DELANY,   gr. 
1881. 


JOLTL.        d 

3-0  /  Ul.  moloM^  ff  cfcLt.  (0)56^ 

/ 

//            // 

"  /OO  60..  &cvu>lina.  RU&  "     7^ 

S'd-.      8 

"    50  0-.  m<Hka,  €^&o    "  281= 

TfloA,.  /7 

"  /  Ul.  ^wfOA,  328  16-.      "     <^i 

a^.S2 

"  /  (My?o  S'&a.  36  U>-.             "  #J^ 

UNITED   STATES   MONEY. 
63. 


Q/^.  ^uTne^    ^oo^ie^. 


113 

New  York,  May  16, 1881, 


TERMS  CASH. 


1Sougl)t  of  ARTHUR  OILMAN  &  CO. 


c^Ot^   ^/eui.  M>il. ®  ^^O.^O. 

SO/.    Q4iiu^  &£e^  Mee/ss         ^T^.^O... 
^a/,   ,^zU  4-^/^ M  ^.//. 


Received  Payment, 


64. 


©^.  'm^/^77z  Ma/Li>. 


Providence,  June  11, 1881. 
So  JOHN  O'BRIEN,  Pt. 


(S/o^c/(zy^'  o^^i @  ^S.^O 

n      ^aUcTia n        ^.yj 


y 


Received  Payment, 

My  M^  0'Muem 


114  UNITED    STATES   MONEY. 

65. 

1S81.  |iOtt5l)t  of  ALBERT  LANE  &  SONS. 


Burlington,  July  19, 1881, 


(3^^ 

"  ^ 

;) 

// 

?i 

/J' 

^u?te 

7 

II 

// 

•  I  ^/ 


66. 

1882.  I^  llCCOttttt  XDltl)   FELIX  REMOND,  pt. 


iV^ei(;  Orleans,  July  1, 1882. 


i^u?ie    (^ 

/^  ///  ©/f  ^^  (^^^^          @   /c^.  J'c^ 

/ 

II          J 

/^  ///(2%  <^^^  (^^.^i  .1       /4^ 

..        ^ 

c^  ^^^  ^u^€ei.,  /J^/  ^         II           .c^^ 

It       Ifj 

/J'^^  (M?zoiec^  (^dcc7?z       II           ./^:? 

.1      J^^ 

4c^  #.    ^/^^ II           .// 

— — 

1.        ^ 

G//i^elcda?zc/ci>e J^^.<^0 

M          /<:^ 

^a^^. OO.OO 

Ma^7zce  c/ue  S^.   M, 

/ 

Received  Payment,  July  3,  1882, 


UNITED    STATES   MONEY.  115 

67.  Trenton,  Oct.  3,  1881.  Andrew  Shaw  bought  of  In- 
gram, Smith,  &  Co.,  25  pairs  of  Kip  Boots  at  $  2.50  a  pair ; 
20  pairs  Calf  Boots  at  $  2.75  a  pair  ;  30  pairs  of  New  Bruns- 
wick Kubber  Boots  at  |  3  a  pair ;  15  pairs  of  Kip  Brogan 
Shoes  at  $  1.25  a  pair ;  and  there  was  charged  for  carting  25 
cents.     Make  out  Mr.  Shaw's  bill  and  find  its  amount. 

.68.  St.  Paul,  Sept.  15,  1881.  Hooper,  Blake,  &  Dudley 
sold  Jacob  Van  Husen  100  bbl.  Best  Test  Patent  Plour  at 
$  8.25  a  barrel ;  50  bbl.  Wilbur's  Extra  Elour  at  I  6.50  a 
barrel ;  and  charged  for  prepaid  freight  $  6.25.  Make  out  the 
bill  and  receipt  it  for  Hooper,  Blake,  &  Dudley. 

69.  Make  out  a  bill,  June  1,  1881,  for  Joseph  Mclntire,  for 
3  months'  services  rendered  him  at  $  28  a  month. 

70.  Chicago,  Nov.  1,  1882.  William  Asbury  owed  George 
W.  Ogden  &  Bro.  for  50  tons  of  Franklin  Coal  at  $  5.25  a 
ton,  bought  Oct.  2 ;  75  tons  of  Cumberland  Coal  at  $  4.75  a 
ton,  bought  Oct.  10  ;  25  cords  of  Pine  Wood  at  $  4.75  a  cord, 
bought  Oct.  16  :  and  George  W.  Ogden  &  Brother  owed  him 
for  cash  paid  Oct.  10,  $  262.50 ;  and  for  bill  of  merchandise 
rendered  Oct.  20,  $  31.65.  Make  out  the  account  and  receipt 
for  the  balance  in  your  name  for  George  W.  Ogden  &  Brother. 

71.  Nov.  10,  1881.  Arthur  Koberts  bought  of  Jordan, 
Marsh,  &  Co.  64 J  yards  of  tapestry  carpet  at  87 J  f  ;  27  yards 
Brussels  carpet  at  $1.85;  18|  yards  oilcloth  at  37^/;  the 
making  and  laying  cost  $13.22.  What  was  the  amount  of 
the  bill  ? 

QUESTIONS. 

148.  Recite  the  table  of  United  States  money.  150.  What  are  thb 
principal  coins  of  the  United  States  ?  154.  Why  are  the  rules  for 
the  processes  in  decimals  applicable  to  processes  in  United  States 
money  ? 

155.  What  are  aliquot  parts  of  a  number  ?  156.  What  are  some 
of  the  aliquot  parts  of  a  dollar  ? 

157.  What  is  an  account?  158.  A  debtor?  A  creditor?  159. 
What  is  a  bill  ?    When  is  a  bill  receipted  ? 


116  WEIGHTS  AND  MEASURES. 


WEIGHTS   AND   MEASURES. 

160.  1.   How  can  the  distance  between  two  places  be  found 
and  expressed  ? 

2.  How  can  you  determine  the  space  occupied  by  a  floor  ? 
The  space  enclosed  by  the  walls  of  a  room  ? 

3.  How  can  you  determine  the  amount  of  water  in  a  pitcher  ? 
The  quantity  of  grain  in  a  bin  ? 

4.  How  can  you  tell  how  much  sugar  you  have  bought  ? 

LENGTH  MEASURES. 

161.  A  Line  is  that  which  has  length  only,  as  the  dis- 
tance between  two  corners  of  this  book. 

162.  Linear,  or  Length  Measures  are  those  used  in  meas- 
uring lines  or  distances. 

TABLE. 

12    inches  (in.)      are  1  foot,  ft. 

3    feet  "  1  yard,  yd. 

5i  yd.,  or  16^  ft.   "  1  rod,  rd. 

320    rd.,  or  5280  ft.  "  1  mile,  mi. 

5.  How  many  inches  in  4  ft.  ?    In  5J  ft.  ?    In  ^  yd.  ?    In 
I  yd.? 

6.  How  many  yards  in  72  ft.  ?    In  90  in.  ?   In  16^  ft.  ?   In 
38  ft.?     In^rd.? 

7.  How  many  rods  in  J  mi.  ?     In  j^  mi.  ?     In  ^  mi.  ?     In 
33  yd.  ? 

8.  At  $  2  a  foot,  what  will  it  cost  to  build  24  rods  of  stone 
wall? 

9.  Required  the  distance  in  yards  around  a  room  13  feet 
square. 

10.  How  many  rods  of  fence  are  required  to  enclose  a  lot 
25  rods  wide  and  twice  as  long  ? 


WEIGHTS   AND   MEASURES. 


117 


11.  In  2  yd.  2  ft.  2  in.,  how  many  inches  ? 

12.  What  cost  27  feet  of  picture-cord  at  10  cents  a  yard  ? 

13.  What  cost  27  in.  of  ribbon  at  60  cents  a  yard  ? 

14.  How  many  inches  in  0.33 J  of  2  yards  ? 


SURFACE  MEASURES. 

163.  A  Surface  is  that  which  has  length  and  breadth 
only,  as  this  page,  or  the  outside  of  a  block. 

A  plane  surface  is  one  that  does  not  change  its  direc- 
tion. 

164.  An  Angle  is  the  difference  in 
direction  of  two  straight  lines  which 
meet  at  a  point,  called  the  vertex  of 
the  angle ;  as  A  D  C  or  C  J)  B.  ^ 


165.  Two  lines  are  perpendicular 
to  each  other  when  they  meet  so  as 
to  form  equal  adjacent  angles.  The 
angles  thus  formed  are  called  Bight 
Angles. 

Thus  the  lines  A  B,C  D  are  perpendicular  to  each  other, 
and  form  the  right  angles,  A  D  C,  C  D  B. 

166.  A  Rectangle  is  a  plane 
surface  with  four  straight  sides 
and  four  right  angles. 


167.  A  Square  is  a  rectangle 
with  equal  sides. 

168.  Surface,  or  Square  Meas- 
ures are  those  used  in  measuring 
surfaces. 


An  Inch  iS^Kore,  or  Square  Inch. 


118 


WEIGHTS   AND   MEASURES. 


TABLE. 

are  I  square  foot,  sq.  ft. 

9    square  feet  "  1  square  yard,  sq.  yd. 

30^  square  yards,  or  2 72  J  sq.  ft.  "  1  square  rod,  sq.  rd. 

160    square  rods  "  1  acre,  A. 

640    acres 

An  acre  contains  43,560  square 


144    square  inches  (sq.  in.) 


"    1  square  mile,  or  section,  sq.  ml 
feet.     A  rood  (R.)  is  40  square  rods. 


ORAL   EXERCISES. 

15.  How  many  square  inclies  in  ^  sq.  ft.  ?     In  f  sq.  ft.  ? 
In  0.5  sq.  ft.  ?     In  \\  sq.  ft.  ? 

16.  How  manj^  square  feet  in  3  sq.  yd.  ?      In  5J  sq.  yd.  ? 
In  200  sq.  in.  ?     In  7  sq.  yd.  8  sq.  ft.  ? 

17.  In  47  surface  feet  how  many  surface  yards  ?    In  80  sq. 
ft.  ?    In  100  sq.  ft.  ? 

18.  How  many  square  rods  in  i  A.  ?  In  y^^-  A.  ?  In  0.75  A.  ? 

19.  What  part  of  an  acre  is  120  sq.  rd.  ?     50  sq.  rd.  ? 


VOLUME    MEASURES. 


169.  A  Solid,  or  Volume,  is 

that  which  has  length,  breadth, 
and  thickness ;  as  a  brick  or  a 
box. 

170.  A  Cube  is  a  solid  bounded 
by  six  equal  squares. 

171.  Cubic  Measures  are  those 
Au  [nch  c«^6e,  01  Cubic  Inch.       ^gg^j  jj^  measuriuff  solids  or  vol- 


umes. 


WEIGHTS   AND   MEASURES.  119 

TABLE. 

1728  cubic  inches  (cu.  in.)    are  1  cubic  foot,  cu.  ft. 
27  cubic  feet  "    1  cubic  yard,  cu.  yd. 

A-lso, 

16  cubic  feet  "    1  cord  foot,  cd.  ft. 

8  cord  feet,  or  128  cu.  ft.  "    1  cord,  cd. 

ORAL    EXERCISES. 

20.  In  3  cu.  yd.  how  many  cubic  feet  ?    In  10 J-  cu.  yd.  ?    In 
81  cu.  ft.  how  many  cubic  yards  ?    In  108  cu.  ft.  ? 

21.  In  4  cd.  how  many  cord  feet  ?    In  5^  cd.  ? 

22.  In  64  cu.  ft.  how  many  cord  feet  ? 

23.  How  many  cord  feet  equal  a  cubic  yard  ? 

CAPACITY  MEASURES. 

172.  Liquid  Measures  are  used  in  measuring  liquids. 

173.  Dry  Measures  are  used  in  measuring  grain,  roots, 
fruit,  etc.  , 

TABLES. 
Liquid.  Dry. 


4  gills  (gi.)  are  1  pint,  pt. 
2  pints  "    1  quart,  qt. 

4  quarts         "    1  gallon,  gal. 


2  pints  (pt.)  are  1  quart,  qt. 
8  quarts  "  1  peck,  pk. 
4  pecks  "    1  bushel,  bu. 


The  gallon  contains  231  cubic  inches  ;  4  quarts  dry  measure,  2684 
cubic  inches  ;  and  the  bushel,  2150.42  cubic  inches. 

A  hogshead  as  a  measure  is  63  gallons  ;  and  a  barrel j  31^  gallons. 

ORAL    EXERCISES. 

24.  How  many  gills  in  5  pt.  ?   In  9J  pt.  ?    In  2  qt.  ?    In  1 
gal.  ?    In  I  qt.  ? 

25.  How  many  quarts  in  7  gal.  ?    In  8J  gal.  ?    In  48  pt.  ? 
In  100  gi.  ?    In  3  gal.  3  qt.  ? 


120 


WEIGHTS   AND   MEASUKES. 


26.  How  many  gallons  in  41  qt.  ?    In  53  qt.  ?     In  60  pt.  ? 
In  120  gi.  ?     In  89  qt.  ? 

27.  How  many  pecks  in  48  quarts  ?     How  many  bushels  ? 

28.  Change  to  larger  measures,  100  pt.  of  milk ;   64  qt.  of 
nuts  ;  84  gi.  of  oil. 

29.  What  is  a  six-gallon  can  of  milk  worth  at  5  cents  a 
quart  ? 

30.  I  take  a  pint  of  milk  at  night  and  a  quart  in  the  morn- 
ing.    What  is  my  weekly  milk-bill  at  7  cents  a  quart  ? 

31.  What  cost  6 1  bu.  of  potatoes  at  $  4  a  huohel  ? 

32.  I  burn  a  pint  of  kerosene  every  night.     What  will  a 
2  weeks'  supply  cost  me  at  20  cents  a  gallon  ? 

33.  Bought  nuts  for  $  2  a  bushel,  and  sold  them  for  10 
cents  a  quart.     How  much  did  I  gain  on  5  bushels  ? 

WEIGHTS. 


174.  Troy  Weights  are  used  in  weighing  gold,  silver, 
and  jewels. 

175.  Avoirdupois  Weights  are  used  in  weighing  all  com- 
mon articles.  , 

TABLES. 
Troy.  Avoirdupois. 


24  grains  (gr.)  are  1  pennyweight. 
20  p^vt.  "    1  ounce,  oz. 

12  ounces  "    1  pound,  lb. 


16  ounces  (oz.)  are  1  pound,  lb. 
100  pounds  "    1  cental,  ctl. 

2000  pounds  "    1  ton,  T. 


1.  The  long  ton  of  2240  avoirdupois  pounds  is  used  at  custom- 
houses, and  in  weighing  coal  at  the  mines. 

2.  A  pound  avoirdupois  is  equal  to  7000  grains,  and  a  pound  troy 
to  5760  grains,  so  that  144  pounds  avoirdupois  are  equal  to  175 
pounds  troy.    A  hundred-weight  (cwt.)  is  the  same  as  the  cental. 


WEIGHTS  AND  MEASURES.  121 


ORAL  EXERCISES. 


34.  In  avoirdupois  weight,  3  lb.  are  how  many  ounces  ?  5^ 
lb.  ?    21  lb.  ?    3  lb.  7  oz.  ?   2  lb.  9  oz.  ?   3f  lb.  ?    .75  lb.  ? 

35.  In  avoirdupois  weight,  56  oz.  are  how  many  pounds  ? 
65  oz.  ?  27  oz.  ?  2500  lb.  are  how  many  tons  ?  4000  lb.  ? 
7000  lb.  ? 

36.  Change  to  the  next  larger  troy  weight,  35  oz. ;  75  oz. ; 
60  pwt. ;  84  pwt. ;  48  gr. ;  30  gr. ;  100  oz. 

37.  Change  to  the  next  smaller  troy  weight,  3  lb. ;  2^  lb. ; 
4  lb.  7  oz. ;  6  oz. ;  5^  oz. ;  2  pwt. ;  2J  pwt. ;  |  lb. ;  f  oz. ;  J 
pwt. ;  0.6  lb. ;  0.8  oz. 

38.  What  cost  2  lb.  4  oz.  of  butter  at  32  cents  a  pound  ? 

39.  What  can  I  get  for  7  pwt.  of  old  silver  at  $1.10 
an  oz.  ? 

40.  How  much  hay  at  $  25  a  ton  can  be  bought  for  $55? 

41.  If  200  lb.  can  be  packed  in  a  barrel,  how  many  barrels 
\¥ill  be  needed  to  pack  2  T.  8  ctl.  ? 


-K 


TIMB. 
176.    Time  is  a  measured  portion  of  duration. 

TABLE. 

60  seconds  (sec.)  are  1  minute,  min. 


Also, 


h   The  Civil  Day  begins  and  ends  at  12  o'clock,  midnight. 


60  minutes 

u 

1  hour,  h. 

24  hours 

a 

1  day,  d. 

365  days 

u 

1  common  year,  y. 

366  days 

u 

1  leap  year,  1.  y. 

7  days 

are  1  week,  wk. 

12  calendar  months 

(mo.)   "    1  year,  y. 

100  years 

"    1  century. 

122  WEIGHTS   AND   MEASURES. 

2.    The  Calendar  Months,  and  the  number  of  days  in  each,  are,  - 


1. 

January, 

31  days. 

7. 

July, 

31  days. 

2. 

February, 

28 

or  29     " 

8. 

August, 

31     " 

3. 

March, 

31     " 

9. 

September, 

30     " 

4. 

April, 

30     " 

10. 

October, 

31     " 

5. 

May, 

31     " 

11. 

November, 

30     " 

6. 

June, 

30     " 

12. 

December, 

31     " 

3.  The  Solar  Year  is  365  d.  5  h.  48  min.  49.7  sec,  or  very  nearly 
365^  days.    The  fraction  in  4  years  amounts  to  nearly  a  day.    Hence, 

When  the  number  of  any  year  is  divisible  by  4  and  not  by  100,  and 
when  it  is  divisible  by  400,  the  month  of  February  has  29  days,  and  the 
year  is  called  a  leap  year. 

ORAL    EXERCISES. 

42.  How  many  minutes  in  2  h.  ?  In  3  J  h.  ?  In  f  h.  ?  In 
3  h.  20  min.  ? 

43.  How  many  hours  in  210  min.  ?     In  3  d.  ?     In  ^  wk.  ? 

44.  How  many  days  in  96  h.  ?     In  60  h.  ? 

45.  How  many  minutes  from  10.25  A.  m.  to  11.45  A.  m.  ? 
From  11.60  A.  m  to  1.35  p.  m.  ? 

46.  What  part  of  a  minute  is  42  sec.  ?  What  part  of  an 
hour  is  48  min.  ? 

47.  How  many  days  from  Mar.  31  to  May  31  ?  From  June 
7  to  July  15  ?     From  Oct.  19  to  Nov.  12  ? 

48.  How  old  is  a  child  on  July  4,  1883,  who  was  born  May 
31,  1880  ? 

49.  If  a  three  weeks'  vacation  begins  June  21,  when  does  it 
end  ?    If  it  ends  Oct.  9,  when  does  it  begin  ? 

50.  What  day  was  25  days  before  Mar.  4,  1880  ?  Sixty  days 
after? 

51.  How  long  from  Thursday  noon  to  4.30  p.  m.  of  Friday  ? 

52.  How  long  is  the  night  when  the  sun  sets  at  7.35  and 
rises  at  4.30  ? 

53.  Name  the  next  six  leap  years. 

54.  What  century  is  this?  In  wb^t  century  did  1620 
come  ? 


WEIGHTS   AND   MEASURES.  123 

ARC    AND    ANGLE    MEASURES. 

177.  A  Circle  is  a  plane  surface 
bounded  by  a  curve,  all  points  of 
which  are  equally  distant  from  a 
point  within  called  the  center. 

The  circumference  of  a  circle  is 
its  boundary  line. 

An  arc  of  a  circle  is  any  part  of 
the  circumference  ;  as  A  C  oi  I)  B. 

The  diameter  of  a  circle  is  a  straight  line  drawn  through 
its  center,  and  terminated  by  the  circumference ;  as  ^  ^. 

The  radius  of  a  circle  is  half  its  diameter ;  ^^AEoiE  D, 

178.  A  Degree  is  -^^  of  any  circumference. 

179.  An  Angle  whose  vertex  is  the  center  of  a  circle 
is  measured  by  the  arc  between  its  sides. 

Thus,  the  arc  A  C measures  the  angle  AEG. 

TABLE. 

60  seconds  (f')  are  1  minute,  \ 
60  minutes        "    1  degree,  ®. 
360  degrees       -  "     1  circumference,  C. 

1.  A  degree  of  the  earth's  equator  contains  69.16  miles,  or  about 
691  miles. 

2.  A  sign  is  an  arc  of  30°  ;  as  D  B.  A  sextant  is  an  arc  of  60°  ;  as 
DC.  A  quadrant  is  an  arc  of  90°  ;  as  ^  C.  A  semi-circumference  is 
an  arc  of  180° ;  as  ^  C^.     A  right  angle  is  an  angle  of  90° ;  as  ^  ^  0. 

Note.  —  The  earth,  by  turning  on  its  axis  once  in  24  hours,  causes  ^  of 
360°,  or  15°,  of  longitude  to  pass  under  the  sun,  from  east  to  west,  in  1  hour's 
time. 

ORAL   EXERCISES. 

55.  How  many  degrees  in  2  right  angles  ?  In  4  right 
angles  ? 

56.  How  many  minutes  in  3^°  ?     In  |°  ? 


124  WEIGHTS   AND   MEASURES. 

57.  In  195'  how  many  degrees  ?    In  360'  ? 

58.  How  many  signs  in  a  quadrant?     Draw  an  angle  of 
45°  ;  another  of  75°. 

59.  How  many  degrees  does  the  minute-hand  of  a  clock 
move  in  3  hours  ?    The  hour-hand  ? 

60.  How  many  degrees  does  the  minute-hand  of  a  watch 
move  in  3 J-  hours  ? 

61.  How  many  degrees  in  half  a  meridian  circle?     How 
many  miles  in  one  such  degree  ? 


MISCELLANEOUS   MEASURES. 


180.    Counting. 

12  ones    are  1  dozen. 
12  dozen   "    1  gross. 
12  gross     "    1  great  gross. 
20  ones      "    1  score. 


181.    Paper. 

24  sheets    are  1  quire. 

20  quires      "  1  ream. 

2  reams      "  1  bundle. 

5  bundles  "  1  bale. 


ORAL    EXERCISES. 

62.  How  many  pencils  in  a  gross  ?    How  many  score  ? 

63.  What  cost  15  doz.  pens  at  $  1  a  gross  ? 

64.  How  many  sheets  in  a  ream  ?    In  J  ream  ? 

65.  What  will  72  sheets  cost  at  15  cts.  a  quire  ? 

66.  If  I  buy  paper  at  20  cents  a  quire,  and  sell  for  a  cent  a 
sheet,  how  much  do  I  gain  on  a  ream  ? 

67.  Name  some  articles  that  are  sold  by  the  gross  ?   By  the 
dozen  ? 

68.  How  much  will  a  great  gross  of  tacks  cost  at  10  cents 
a  paper  ? 

69.  Bought  3  gross  of  pens  at  $  1.50  a  gross,  and  sold  them 
for  2  cents  each  j  liow  much  did  I  gain  ? 


WEIGHTS   AND   MEASURES.  125 

70.  How  many  dozen  pint  bottles  will  be  needed  to  hold 
5  gal.  3  qt.  of  currant  wine  ? 

71.  If  I  use  1  ream  of  paper  in  10  weeks,  how  many  sheets 
do  I  use  per  day  ? 

72.  If  you  can  count  four  score  in  a  minute,  how  many  can 
you  count  between  8.50  and  9.10,  A.  m.  ? 

73.  How  many  gross  of  boxes  at  2/  each  can  be  bought  for 
$5.76? 

74.  If  the  sun  requires  24  hours  to  make  his  apparent 
journey  around  the  earth,  how  many  degrees  does  he  travel  in 
an  hour  ? 

75.  A  druggist  buys  a  cask  of  liquor  containing  30  gallons 
for  $  30,  and  sells  it  at  $  0.75  a  pint.     What  does  he  gain  ? 

QUESTIONS. 

161.  What  is  a  line  ?  162.  What  are  linear  measures  ?  Eecite  the 
table  of  linear  measures. 

163.  What  is  a  surface  ?  A  plane  surface?  164.  What  is  an  an- 
gle ?  165.  When  are  two  lines  perpendicular  to  each  other  1  166. 
What  is  a  rectangle  ?  167.  A  square  ?  168.  For  what  are  surface, 
or  square  measures  used  ?     Give  the  table. 

169.  What  is  a  solid,  or  volume?  170.  What  is  a  cube?  171. 
What  are  cubic  measures  ?     Give  the  table. 

172.  For  what  are  liquid  measures  used  ?  Give  the  tables.  173. 
What  articles  are  measured  by  the  bushel  ? 

174.  For  what  are  troy  weights  used  ?  Give  the  table.  175.  For 
what  are  avoirdupois  weights  used  ?  Give  the  table.  What  is  the 
long  ton?  Which  is  the  heavier,  a  pound  of  gold  or  a  pound  of 
sugar  ? 

176.  What  is  time?  Kecite  the  table.  When  does  the  civil  day 
begin  and  end  ?  Name  the  calendar  months.  When  has  February 
29  days  ? 

177.  What  is  a  circle  ?  The  circumference  of  a  circle  ?  The  di- 
ameter ?  178.  What  is  a  degree  ?  179.  How  is  an  angle  measured  ? 
Give  the  table.  What  is  a  degree  at  the  equator  ?  What  is  a  sign  ? 
A  right  angle  ? 

180.  Give  the  counting  table.     181.  Give  the  table  for  paper. 


126  COMPOUND   NUMBERS. 


COMPOUND    NUMBERS. 

182.  1.   How  many  inches  in  3  feet  6  inches  ? 

2.  How  many  feet  and  inches  in  42  inches  ? 

3.  How  many  quarts  in  6  gallons  3  quarts  ? 

4.  How  many  gallons  and  quarts  in  27  quarts  ? 

183.  A  Denomination  is  the  name  of  a  unit  of  measure  oi 

weight. 

184.  A  Denominate  Number  is  a  number  composed  of 
units  of  one  or  more  denominations.     Thus, 

42  inches,  6  pounds  4  ounces,  are  denominate  numbers. 

185.  A  Simple  Number  is  a  number  of  a  single  kind  or 
denomination.     Thus, 

2,  $  3,  5  months,  7  yards,  are  simple  numbers. 

186.  A  Compound  Number  is  a  number  composed  of  two 
or  more  denominations  of  the  same  general  kind.     Thus, 

3  ft.  6  in.,  6  gal.  3  qt.,  are  compound  numbers. 

REDUCTION. 
To  change  Denominate  Numbers  to  Smaller  Denominations. 

ORAL    EXERCISES. 

5.  How  many  inches  in  7  feet  ?     In  5  feet  6  inches  ? 

6.  How  many  gills  in  5  gallons  ?     In  2  gal.  3  qt.  ? 

7.  How  many  ounces  in  3  lb.  troy  ?     In  8  lb.  7  oz.  ? 

8.  How  many  minutes  in  4  h.  20  min.  ?     In  5  h.  15  min.  ? 

9.  How  many  quires  of  paper  in  10  reams  ?     How  many 
sheets  in  3  quires  16  sheets  ? 


COMPOUND   NUMBERS.  127 

10.  How  many  days  in  12  weeks  6  days  ?      In  11  weeks 
4  days  ? 

11.  How  many  quarts  in  3  pecks  ?     In  2  bu.  1  pk.  ? 

187.  Reduction  Descending  is  changing  denominate  num- 
bers to  smaller  denominations. 

WRITTEN    EXERCISES. 

12.  How  many  inches  in  124  rd.  4  yd.  2  ft.  ? 

124  (rd.)  4  yd.  2  ft.  ^^^^^  .^^^  _  1  rd.  =  5^  yd.  ;  124 

^  J^'  rd.  =  124  X  H  yd.,  or  682  yd., 

62  which,  with  4  yd.  added,  are  686 

620  yd. 

^  .    . .  1  yd.  3=  3  ft.  ;  686  yd.  =  686  X 

DOD  (^ya.;  3  ^^^  ^^^  ^058  ft.,  which,  with  2  ft. 

_?  ^*-  added,  are  2060  ft. 


2060  (ft.)  1  ft.  =  12  in.  ;  2060  ft.  3=  2060 

12  in.  X  12  in.,  or  24720  in.      124  rd.  4 

OATOf)  *  yd-  ^  ^^'  ^^^  24720  inches. 

13.  How  many  pints  in  19  bu.  3  pk.  7  qt.  1  pt.  ? 

14.  How  many  cubic  feet  in  48  cu.  yd.  15  cu.  ft.  ? 

15.  How  many  pounds  in  17  T.  17  cwt.  90  lb.  ? 

188.    Rule  for  Reduction  Descending. 

Multiply  the  largest  denomination  by  the  number  of  units  it 
takes  of  the  next  smaller  denomination  to  equal  one  of  that 
larger,  and  to  the  product  add  the  given  number,  if  any,  of 
the  smaller  denomination. 

Multiply  the  sum  in  like  manner,  and  so  proceed  until  the 
given  number  is  changed  to  units  of  the  required  denomina- 
tion. 

16.  Eeduce  12  A.  144  sq.  rd.  144  sq.  ft.  to  square  feet. 

17.  Eeduce  5  cu.  yd.  23  cu.  ft.  725  cu.  in.  to  cubic  inches. 

18.  Reduce  60  gal.  3  qt.  1  pt.  to  pints. 


+ 


128  COMPOUND    NUMBERS. 

19.  Eeduce  13  bii.  2  pk.  7  qt.  1  pt.  to  pints. 

20.  Reduce  47  miles  to  feet. 

21.  Eeduce  72  lb.  10  oz.  15  pwt.  7  gr.  to  grains. 
22  Eeduce  43  T.  13  cwt.  20  lb.  to  pounds. 

23.  Eeduce  365  d.  5  h.  48  min.  50  sec.  to  seconds. 

24.  How  many  cubic  inches  in  8  cu.  yd.  10  cu.  ft.  728  cUo  in.? 

25.  How  many  seconds  in  45°  28'  54^'  ? 

26.  How  many  cubic  feet  in  25  cd.  ? 

27.  How  many  gills  in  40  gal.  3  qt.  0  pt.  2  gi.  ? 

28.  How  many  minutes  in  67  wk.  6  d.  9  h.  52  min.  ? 

29.  How  many  sheets  of  paper  in  4  bundles  1   ream   of 
paper  ? 

30.  What  is  the  value  of  1  A.  80  sq.  rd.  of  land  at  5  cents  a 
square  foot  ? 

31.  What  will  be  the  cost  of  grading  3  m.  195  rd.  of  road 
at  1 6.50  a  rod  ? 

32.  Change  ^  of  a  mile  to  feet. 

660 
Solution.  —  1  mile  =  5280  ft.  ;  J  mi.  =  -J  of  5280  ft.,  or  |  X  ^^^jH  ft. 
=  4620  ft. 

33.  Eeduce  ^g-  of  a  mile  to  yards. 

34.  What  part  of  a  grain  is  -g-^^jj  of  a  pound  troy  ? 

35.  What  part  of  a  second  is  s^^jjjj  of  a  day  ? 

36.  What  part  of  a  gill  is  -z^^-g  of  a  gallon  ? 

37.  Eeduce  ^  oi  o,  bushel  to  quarts. 
3a    Eeduce  1.375  gallons  to  pints. 

1.375  (gal.) 

4  q^^  Solution.  —  1  gal.  =  4qt. ;  1.375  gal.  =  1.375 

rKAA  /  i.  \  X  4  qt.,  or  6.5  qt. 

^^^'^  1  qt.  =  2  pt.  ;  6.5  qt.  =  5.5  X  2  pt.,  or  1 1 

t.V^-  pints. 

11.000  pt. 


COMPOUND   NUMBERS.  129 

39.  Reduce  .024  of  a- ton  to  pounds. 

40.  Reduce  .0075  of  an  acre  to  square  feet. 

41.  Eeduce  .3945  of  a  day  to  minutes. 

42.  How  many  square  yards  in  1.364  acres  ? 

To  change  Denominate  Numbers  to  Larger  Denominations. 
ORAL   EXERCISES. 

43.  How  many  feet  in  84  inches  ?     In  66  inches  ? 

44.  How  many  gallons  in  160  gills  ?     In  88  gills  ? 

45.  How  many  pounds  troy  in  36  ounces  ?    In  103  ounces  ? 

46.  How  many  hours  in  260  minutes  ?     In  315  minutes  ? 

47.  How  many  reams  in  200  quires  of  paper  ?     In  76  sheets 
of  paper  how  many  quires  ? 

48.  How  many  weeks  in  90  days  ?     In  81  days  ? 

49.  How  many  pecks  in  24  quarts  ?     In  72  quarts  ? 

189.    Reduction  Ascending  is  changing  denominate  num- 
bers to  larger  denominations. 

WRITTEN    EXERCISES. 

50.  Change  410  feet  to  larger  denominations. 


3    ft. 
5iyd. 

2 


4^0  fi;^  Solution.  — -  As  in 

136^.  +2  ft  ■  3  feet  there  is  1  yard, 

•^  there   are   as    many 


11  hf  .-yd, 


_  yards  in  410  feet  as 

272  hf.-yd.  times  3  feet  in  410 


24  rd.  +  8  hf.-yd.,  or  4  yd.        feet,  which  are  136, 

and  2  feet  remainder. 
As  in  5^  yards  there  is  1  rod,  there  are  as  many  rods  in  136  yards  as 
times  5^  yards  in  136  yards,  or  times  11  half-yards  in  272  half-yards, 
or  12,  and  8  half-yards,  or  4  yards,  remainder. 
In  410  feet  there  are  24  rd.  4  yd.  2  ft. 

51.  Change  1279  dry  pints  to  larger  denominations. 

52.  Change  1311  cubic  feet  to  larger  denominations. 

53.  Change  35790  pounds  to  larger  denominations. 

9 


^ 


x 


130  A         COMPOUND  numbehs. 

190.    Rule  for  Reduction  Ascending. 

Divide  the  given  number  by  the  number  of  units  it  takes  oj 
that  denomination  to  equal  one  of  the  next  larger,  and  reserve 
the  remainder,  if  any*  Divide  the  quotient  in  like  manner, 
and  so  proceed  until  the  required  denomination  is  reached. 

The  last  quotient,  and  the  several  remainders,  if  any,  will 
be  the  number  sought. 

Note.  —  Reduction  Ascending  and  Reduction  Descending,  being  reverse 
processes,  are  proofs  of  each  other. 

Change  to  larger  denominations : 

54.  562068  square  feet.  61.   31556930  seconds. 

55.  273749  cubic  inches.  62.  391256  cubic  inches. 

56.  487  liquid  pints.  63.   163734''. 

57.  879  dry  pints.  64. .  3200  cubic  feet. 

58.  248160  feet.  65.   1306  gills. 

59.  419887  grains.  66.   684592  minutes. 

60.  87320  pounds.  67.   4320  sheets  of  paper. 

68.  At  5  cents  a  square  foot,  how  many  acres  and  square 
rods  of  land  can  be  bought  for  $  3267  ? 

69.  At  $  5.50  a  rod,  how  many  miles  and  rods  of  road  can 
be  graded  for  %  6352.50  ? 

70.  Change  3080  ft.  to  the  fraction  of  a  mile. 

Solution.  —  1  mi.  =  5280  ft.  ;  3080  ft. 
mi.  =  -j^j  mi.  are  |§|^  mi.,  which,  changed  to  sma^le^t 

terms,  is  ^  mi. 


■f 


71.  Reduce  88  yards  to  a  fraction  of  a  mile. 

72.  What  part  of  a  pound  troy  are  3000  grains  ? 

73.  What  part  of  a  day  are  12600  seconds  ? 

74.  What  part  of  a  gallon  are  24  gills  ? 

75.  Reduce  4^^  quarts  to  a  fraction  of  a  bushel. 


COMPOUND   NUMBERS.  131 

76.    Reduce  11  pints  to  a  decimal  of  a  gallon. 

2)  11.000  pt.  Solution.  —  As  in  1  quart  there  are  2  pints, 

4")    5  500  at.  there  are  in  11  pints  ^  as  many  quarts,  or 

^     - — ~  11  ~  2  =  5.5  quarts. 

l.o  /  o  ga  .  As  in  1  gallon  there  are  4  quarts,  there  are 

in  5.5  quarts  \  as  many  gallons,  or  5.5  -~  4  =  1.375  gallons. 

77  Reduce  48  pounds  to  a  decimal  of  a  ton. 

78.  Reduce  326.7  square  feet  to  a  decimal  of  an  acre. 

79.  What  decimal  of  a  day  is  568.08  minutes  ? 

80.  How  many  acres  in  6601.76  square  yards  ? 

To  change  Denominate  Fractions  to  Integers  of  Smaller 
Denominations. 

ORAL  EXERCISES. 

81.  How  many  quarts  in  |  of  a  peck  ? 

82.  How  many  hours  in  ^  of  a  day  ? 

83.  How  many  feet  and  inches  in  f  of  a  yard  ? 

Solution.  —  f  of  a  yard  =  |  of  3  feet  =  2^  feet ;  ^  of  a  foot  =  ^  of 
12  inches  =  6  inches  ;  f  of  a  yard  =  2  feet  6  inches. 

84.  How  many  days  and  hours  in  |  of  a  week  ? 

85.  How  many  ounces  and  pennyweights  in  .8  of  a  pound  ? 

86.  How  many  quarts  and  pints  in  .6  of  a  gallon  ? 

191.    A  Denominate  Fraction  is  a  fraction  of  a  denomi- 
nate number. 

WRITTEN   EXERCISES. 

87.  Eeduce  ^  of  a  bushel  to  units  of  smaller  denominations. 

r^  bu.  =  I  of  4  pk.  =  3^  pk. 

Solution.  K  ^  pk.  =  I  of  8  qt.  =  Of  qt. 

(f  qt.  =f  of2pt.  =l^pt. 

Ana.  3pk.  Oqt.  l^^pt. 


132  COMPOUND   NUMBERS. 

88.  Keduce  §  of  a  mile  to  units  of  smaller  denominations. 

89.  What  is  tlie  value  of  .83  of  a  bushel  in  smaller  denom- 
inations ? 

First  Solution,  Second  Solution. 

.83  bu.  =  .83  of  4  pk.  =  3.32  pk.  .83  bii. 

.32  pk.  =  .32  of  8  qt.   =  2.56  qt.  f 

^6  qt.  =  .56  of  2  pt.  =  1.12  pt.  3.32  pk. 

8 

Ans.  3  pk.  2  qt.  1.12  pt.  2.56  qt. 

2 

1.12  pt. 

90.  Keduce  .53  rods  to  units  of  smaller  denominations. 

192.    Rule  for  Reduction  of  Denominate  Fractions  to  Integers. 

Change  the  fraction^  as  far  as  possible,  to  an  integer  of  the 
denomination  next  smaller.  If  there  is  a  fraction  in  the  re- 
sult, proceed  with  it  in  like  manner,  and  so  continue  as  far 
as  required. 

Eeduce  to  integers  of  smaller  denominations  : 

91.  f  of  an  acre.  J^  95.    .6725  of  a  cental. 

92.  I  of  a  pound  troy.  96.    .282  of  a  ton. 

93.  -/y  of  a  common  year.      97.    .875  of  a  rod. 

94.  ^  of  a  mile.  98.   .761  of  a  day. 

To   change   Denominate   Integers  to  Fractions  of  Larger 
D  enominations. 

ORAL    EXERCISES. 

99.   What  part  of  a  peck  is  6  quarts  ? 

100.  What  part  of  a  day  is  21  hours  ? 

101.  What  fraction  of  a  pound  is  14  ounces  ? 

102.  What  fraction  of  a  yard  is  2  feet  6  inches  ? 

103.  What  fraction  of  a  gallon  is  2  quarts  1  pint  ? 


COMPOUND    NUMBEKS.  133 

104.  What  decimal  of  a  bushel  is  3  pecks  ? 

105.  What  decimal  of  an  ounce  is  16  pennyweights  ? 


WRITTEN    EXERCISES. 

106.   What  fraction  of  a  bushel  is  3  pk.  0  qt.  1 J  pt.  ? 

rHpt.  =:^/pt.=:V--2,orfqt. 
Solution.  -(  Of  qt.  =  f  -^  8,  or  ^  pk. 

(  3i  pk.  =  ^-  pk.  =  2^-  ^  4,  or  ^  bu.,  Ans. 


107.  What  part  of  a  mile  is  71  rd.  1  ft.  10  in.  ? 

108.  What  decimal  of  a  bushel  is  3  pk.  2  qt.  1.12  pt.  ? 

r  1.12  pt.  =  1.12  -^  2  ==  .56  qt.  ^  2)  1.12  pt. 

Solution.  J  2.56  qt.  =  2.56  -^  8  =  .32  pk.  I  Or,  8)  2^  qt. 

(  3.32  pk.  =  3.32  -^  4  =:  .83  bu.  J  4)  3^  pk. 

Ans.  .83  bu.  .83  bu. 

109.  Eeduce  2  yd.  2  ft.  8.94  in.  to  a  decimal  of  a  rod. 

193.    Rule  for  Reduction  of  Integers  to  a  Denominate  Fraction. 

Change  the  given  number  of  the  smallest  denomination  to  a 
fraction  of  the  next  larger.  Write  this  fraction  as  a  part  of 
the  number  of  that  larger  denomination.  Change,  in  like 
manner,  the  number  thus  formed,  and  so  proceed  as  far  as 
required. 

110.  Eeduce  68  sq.  rd.  155  sq.  ft.  to  a  fraction  of  an  acre. 

111.  Eeduce  10  oz.  13  pwt.  8  gr.  to  a  fraction  of  a  pound. 

112.  Eeduce  232  d.  10  h.  21  min.  to  a  fraction  of  a  year. 

113.  Eeduce  248  rd.  4  yd.  2  ft.  8  in.  to  a  fraction  of  a  mile. 

114.  Eeduce  67  lb.  4  oz.  to  a  decimal  of  a  cental. 

115.  Eeduce  5  cwt.  64  lb.  to  a  decimal  of  a  ton. 

116.  Eeduce  4  yd.  2  ft.  5.25  in.  to  a  decimal  of  a  rod. 

117.  Eeduce  18  h.  15  min.  50.4  sec.  to  a  decimal  of  a  day. 


134  COMPOUND    NUMBERS. 

194.  When  it  is  required  to  find  the  part  that  one  de- 
nominate number  is  of  another, 

Reduce  the  numbers  to  the  same  denomination,  and  divide 
the  result  denoting  the  part  hy  that  denoting  the  whole, 

.    118.   What  part  of  2  A.  112  sq.  rd.  is  144  sq.  rd  ? 

119.  What  part  of  3  mi.  120  rd.  4  yd.  is  2  mi.  120  rd.  3  yd.  \ 

120.  What  part  of  1  lb.  4  oz.  12  pwt.  12  gr.  is  5  oz.  10  pwt.  ? 

121.  What  decimal  of  74  mi.  80  rd.  is  9  mi.  90  rd.  ? 

122.  What  decimal  of  7  bu.  1  pk.  5  qt.  is  82  bu.  3  pk.  1  qt.  ? 


V 


ADDITION. 


195.  The  operations  with  compound  numbers  differ 
from  those  with  simple  numbers  only  in  the  fact  that 
compound  numbers  have  an  irregular  instead  of  a  decimal 
scale.  * 

The  principles,  however,  being  the  same,  no  special  rules 
are  required  for  adding,  subtracting,  multiplying,  or  divid- 
ing compound  numbers. 

WRITTEN    EXERCISES. 

123.  Add  15  lb.  11  oz.  19  pwt.  22  gr.,  71  lb.  10  oz.  13  pwt. 
17  gr.,  and  ^h  lb.  9  oz.  17  pwt.  14  gr. 

15  lb.  11  oz.  19  pwt.  22  gr.  Solution.  —  The  sum  of  the 

71        10         13  17  grains  is  53  gr.  =  2  pwt.  5  gr. 

QK         9        17  14  ^^  write  the  5  gr.  beneath, 

r^jT— ; and  add  the  2  pwt.  with  the 

153  lb.    8  oz.  11  pwt.    5  gr.  .  ,  ^ 

^  ^  pennyweights. 

The  sum  of  the  pennyweights  is  51  pwt.  =  2  oz.  11  pwt.  We  write 
the  1 1  pwt.  beneath,  and  add  the  2  oz.  with  the  ounces. 

The  sum  of  the  ounces  is  32  oz.  =  2  lb.  8  oz.  We  write  the  8  oz. 
beneath,  and  add  the  2  lb.  with  the  poundrt. 

The  sum  of  the  pounds  is  153  lb.,  wliich  we  write  beneath,  and 
have,  as  the  entire  sum,  153  lb.  8  oz.  11  pwt.  6  gr. 


COMPOUND   NUMBERS.  135 


124. 

125. 

mi. 

rd. 

ft.  in. 

A. 

sq.  rd. 

sq.  ft. 

sq.  in, 

21 

296 

11   1 

65 

169 

272 

143 

46 

279 

10  11 

80 

134 

260 

116 

35 

214 

9   9 

14 

110 

166 

135 

68 

276 

16  10 

66 

68 

177 

131 

64 

70 

16   1 

60 

161 

69 

117 

216 

177 

I'ti  8 

278 

136 

131i 

66 

Or, 

Or, 

216 

177 

15   2 

278 

136 

131 

102 

In  Ex.  124,  the  ^  ft.  in  the  sum,  reduced  to  units  of  a  lower  denom- 
ination, gives  6  in.,  which  added  with  the  8  in.  of  the  sum,  and  re- 
duced, gives  the  second  form  of  answer.  In  Ex.  125,  the  -^  sq.  ft.,  in 
like  manner,  gives  36  sq.  in.,  which  added  to  the  66  sq.  in.  of  the 
sum,  and  reduced,  gives  the  second  form  of  answer. 

126.  What  is  the  sum  of  14  cu.  yd.  20  cu.  ft.  1463  cu.  in., 
9  cu.  yd.  20  cu.  ft.  1463  cu.  in.,  11  cu.  yd.  23  cu.  ft.  67  cu.  in., 
and  27  cu.  yd.  1305  cu.  in.  ? 

127.  Find  the  sum  of  18  gal.  3  qt.,  60  gal.  3  qt.  1  pt.,  61  gal. 
3  qt.,  and  57  gal.  3  qt.  1  pt. 

128.  Find  the  sum  of  15  d.  23  h.  55  min.  17  sec,  13  d.  15  h. 
17  min.  38  sec,  10  d.  23  h.  42  min.  17  sec,  16  d.  16  h.  38  min. 
47  sec,  and  20  d.  52  min.  57  sec. 

129.  What  is  the  sum  of  28°  56'  58'',  21°  51'  37",  and  13° 
39'  57"  ? 

130.  Add  .761  d.  4-1-  h.  and  1  h.  8  min.  31  sec. 

761  d.  =  18  h.  15  min.  50.4  sec.  Solution.  —  .761     d. 

4j  h.  =   4      15  0  ^^^  ^i  ^'  ^^^  ^^^^  ^^' 

j[         g  3^  duced  to  units  of  small- 

■ er    denominations,    and 

23  h.  39  min.  21.4  sec  ^jj^^  .^ith  the  1  h.  18 

min.  31  sec,  give,  as  the  answer,  23  h.  39  min.  21.4  sec. 


136  COMPOUND    NUMBERS. 

131.  Find  tlie  sum  of  ^^  of  a  ton,  \\  of  a  cental,  and  1  T. 
2  cwt.  3  lb. 

132.  A  man  traveled  one  day  60j  miles,  the  second  day 
50  mi.  120  rd.,  and  the  third  day  h^"^  miles.  How  far  did  he 
travel  in  the  three  days  ? 

,  133.  I  have  three  lots,  —  the  first  contains  f  of  an  acre,  the 
second  §  of  an  acre,  and  the  third  |J  of  an  acre.  How  many 
acres  have  I  ? 

SUBTRACTION. 

134.  Find  the  difference  between  15  lb.  3  oz.  12  pwt.,  and 
9  lb.  1  oz.  17  pwt. 

15  lb.  3  oz.  12  pwt.  Solution.  — As  17  pwt.  cannot  be  taken 

9        1         17  from  12  pwt.,  we  take  1  oz.,  =  20  pwt., 

■"T;     I         ~  from  the  3oz.,  leaving  2oz.,  and  add 

6  lb.  1  oz.  15  pwt.  +     ^1,     in       /      I.'  ^    •        owT 

^  to  the   12  pwt.,  which  gives  32  pwt.  ; 

17  pwt.  taken  from  32  pwt.  leaves  15  pwt.     We  write  the  15  pwt. 

beneath. 

1  oz.  from  2oz.  leaves  1  oz.,  which  we  write  beneath;  9  lb.  from 

15  lb.  leaves  6  lb.     Ans.   6  lb.  1  oz.  15  pwt. 

135.  136. 

From       73  bu.  2  p.  5  qt.  17  mi.  311  rd.  1  yd.  1  ft.  3  in. 

Take        59        3      7  3  79       1        2       7 


137.  From  116  A.  53  sq.  rd.  100  ft.  113  in.  take  87  A.  137 
&q.  rd.  100  sq.  ft.  113  sq.  in. 

138.  The  longitude  of  Boston  is  71^  4^  9'^  W.  and  that  of 
Chicago  87°  35'  W.  What  is  the  difference  in  the  longitude 
of  the  two  places  ? 

139.  From  {^  of  a  pound  troy  take  2  oz.  19.2  gr. 

140.  From  a  hogshead  of  molasses  J  had  leaked  out.  How 
much  remained  ? 

141.  From  .367  of  a  year  take  .761  of  a  day. 

142.  The  distance  between  two  places  is  .7895  of  a  mila 
How  much  is  that  more  than  }  of  a  mile  ? 


COMPOUND   NUMBEliSo  137 

196.      To  find  the  Time  between  two  Dates. 

143.  What  is  the  time  from  May  16, 1819,  to  April  9, 1881  ? 
1881  3  mo.  9  d.  Solution.  —  Of  the  year  1881,  3  mo.  9  d 
1819  4  16  and  of  the  year  1819,  4  mo.  16  d.  have  passed 
at  the  time  named.     As  16  d.  cannot  be  taken 

61  10  24  from  9  d.  we  take  the  3d  month,  March,  hav- 

ing 31  d.  and  add  it  to  the  9  d.,  making  40  d.  and  40  d.  less  16  d.  = 
24  d.  1  y.,  or  12  mo.,  added  to  the  2  mo.  remaining  in  the  minuend 
=  14  mo.,  and  14  mo.  less  4  mo.  =:  10  mo.  1880  y.  less  1819  y.  = 
61  y.     Hence  the  difference  between  the  dates  is  61  y.  10  mo.  24  d. 

Note.  —  The  calendar  month  taken  to  add  to  the  days  is  the  month 
preceding  the  one  named  in  the  later  date,  and  the  number  thus 
added  must  always  be  the  number  of  days  in  the  month  taken  — 
that  is,  30  d.  for  Apr.,  June,  Sept.,  or  Nov.,  28  or  29  d.  for  Feb.,  and 
31  d.  for  any  other  month. 

144.  "Wliat  is  tlie  time  l)etween  Oct.  16,  1876,  and  Aug.  9, 
1882? 

145.  A  note  was  given  Kov.  15, 1879,  and  paid  July  5, 1881. 
How  long  did  it  run  ? 

146.  How  long  from  the  surrender  of  Cornwallis,  Oct.  19, 
1781,  to  the  battle  of  New  Orleans,  Jan.  8,  1815  ? 

147.  What  time  elapsed  from  the  declaration  of  American 
independence,  July  4,  1776,  to  the  emancipation  proclamation, 
Jan.  1,  1863  ? 

148.  How  many  days  are  there  from  Dec.  19,  1879,  to  Mar. 
16,  1880  ? 

12  4-  31  4-  29  4-  16  =  88  Solution.  — There  are  12  days 

remaining  in  December,  31  in 
January,  29  in  February,  and  16  in  March,  or,  in  all,  88  days. 

149.  A  note  dated  April  9  is  to  be  paid  June  8,  1881. 
How  long  has  it  to  run  ? 

150.  Find  the  exact  time,  in  days,  between  May  25,  1880, 
10  p.  M.,  and  March  4,  1881,  9  A.  m. 


138  COMPOUND   NUMBERS. 

MULTIPLICATION. 

151.  Multiply  11  bu.  3  pk.  2  qt.  by  7. 

11}^      ^    Ir  9    f  Solution.  —  7  times  2  qt.  are  14  qt.  = 

1  pk.  and  6  qt.     We  write  the  6  qt.  uii- 

der  the  quarts,  and  reserve  the  1  pk.  to 

82  bu.  2  pk.  6  qt.  add  to  the  7  times  3  pk. 

7  times  3  pk.  =21  pk.,  which  plus  the 
I  pk.  reserved  =  22  pk.  =  5  bu.  and  2  pk.  We  write  the  2  pk. 
under  the  pecks,  and  reserve  the  5  bu.  to  add  to  the  7  times  11  bu. 

7  times  11  bu.  =  77  bu.,  which  plus  the  5  bu.  reserved  =  82  bu. 
Ans.   82  bu.  2  pk.  6  qt. 

152.  Multiply  17  wk.  4  d.  23  h.  47  min.  by  8. 

153.  Multiply  3  mi.  40  rd.  4  yd.  2  ft.  by  12. 

154.  If  1  load  of  hay  weighs  1  T.  3  cwt.  17  lb.,  what  will 
9  loads  weigh  ? 

155.  How    much    land    in   14   farms    of   25  A.   60  sq.  rd. 
21  sq.  yd.  each  ? 

156.  If  the  moon  moves  in  her  orbit  13°  11'  35^'  in  1  day, 
how  far  will  she  move  in  20  days  ? 

DIVISION. 

157.  Find  1  of  82  bu.  2  pk.  6  qt. 

7\  89  V.      9    V   fi    f  Solution. —  4"  ^^  82bu.  =  11  bu., 

^ ! £_ i_l  with  a  remainder  of  5  bu.     The  5  bu. 

11  bu.  3  pk.  2  qt.  =  20  pk.,   which  added  to  the  2  pk. 

in  the  dividend  =  22  pk.  |  of  22  pk. 
^=  3  pk.,  with  a  remainder  of  1  pk.  The  1  pk.  =  8  qt,  which  added 
to  the  6  qt.  =  14  qt.     -f  of  14  qt.  =  2  qt.     Ans.  11  bu.  3  pk.  2  qt. 

isa   Divide  139  wk.  6  d.  10  h.  16  min.  by  8. 
159.   Divide  40  mi.  210  rd.  5  yd.  2  ft.  by  12. 
160    If  12  spoons  weigh  31b.  10  oz.  11  pwt,  what  is  the 
weight  of  each  spoon  ? 


COMPOUND   NUMBERS.  139 

161.  What  is  the  daily  motion  of  the  moon,  if  it  moves 
197°  38'  4:5''  in  15  days  ? 

162.  A  planter  has  1634  gal.  1  qt.  1  pt.  of  molasses,  which 
he  wishes  to  put  into  25  equal  casks.  What  must  be  the  least 
capacity  of  each  cask  to  exactly  receive  the  molasses  ? 

MISCELLANEOUS    EXERCISES. 

163.  How  many  inches  in  18  rd.  5  yd.  2  ft.  11  in.  ? 

164.  What  will  5  T.  17  cwt.  25  lb.  of  iron  cost  at  3  cents  a 
pound  ? 

165.  At  3  cents  a  pound,  how  many  tons  of  iron  can  be 
bought  for  $396.18? 

166.  What  fraction  of  a  common  year  is  27  of  a  day  ? 

167.  In  one  barge  there  are  50  T.  5  ctl.  75  lb.,  and  in  another 
47  T.  17  ctl.  35  lb.     How  many  tons  in  both  barges  ? 

168.  Two  boats  start  in  a  race,  and  one  of  them  gains  5  feet 
upon  the  other  in  every  55  yards.  How  many  rods  will  it 
have  gained  at  the  end  of  2  miles  ? 

169.  If  2  A.  65  sq.  rd.  can  be  plowed  in  a  day,  how  much 
can  be  plowed  in  8J  days  ? 

170.  Find  the  exact  number  of  days  from  June  11,  1879,  to 
Aug.  5,  1881. 

171.  If  9  acres  produce  21 T.  537  lb.  of  hay,  what  does  one 
acre  produce  ? 

172.  If  a  pendulum  vibrates  47  times  in  a  minute,  in  what 
time  will  it  vibrate  13267583  times  ? 

173.  What  decimal  of  20  acres  is  7  A.  148  sq.  rd.  ? 

174.  If  a  man  can  cut  24  cords  102  cubic  feet  of  wood  in 
12  days,  how  many  cord  feet  can  he  cut  in  one  day  ? 

175.  How  many  silver  spoons,  each  weighing  2  oz.  10  pwt., 
ann  be  made  from  a  bar  of  silver  weighing  11  lb.  5  oz.  10  pwt.  ? 


140  COMPOUND    NUMBERS. 

176.  A  stationer  buys  25  reams  of  commercial  note  paper 
at  $  1.75  a  ream,  and  retails  it  at  12  cents  a  quire,  with  tlie 
exception  of  one  outside  quire  of  each  ream,  which  he  sells  at 
8  cents.     How  much  does  he  make  ? 

177.  How  many  years,  months,  and  days  did  a  man  live 
who  was  born  March  15,  1767,  and  died  June  8,  1845  ? 

178.  Bought  a  cask  of  oil,  containing  6S^  gallons,  at  72 
cents  a  gallon  ;  |  having  leaked  out,  the  remainder  was  sold 
at  90  cents  a  gallon.     Did  I  make  or  lose,  and  how  much  ? 

179.  Show  that  any  article  is  worth  as  many  five-cent  pieces 
a  cental  as  dollars  a  ton. 

180.  A  carpenter  sent  two  of  his  apprentices  to  ascertain 
the  length  of  a  certain  fence.  The  first  made  it  17  rd.  16  ft. 
11  in.,  and  the  second  made  it  18  rd.  5  in.  The  carpenter, 
fearing  they  might  both  be  wrong,  measured  for  himself,  and 
found  it  to  be  17  rd.  5  yd.  1  ft.  11  in.  What  was  the  differ- 
ence in  their  measurements  ? 

181.  If  A  and  B  should  commence,  March  5,  1882,  to  go  to 
bed  at  the  same  hour,  and  A  should  rise  at  ^  before  6  o'clock 
and  B  at  f  past  7,  how  much  more  time  for  labor  would  A 
have  had  than  B,  by  March  5,  1900,  paying  attention  to  the 
leap  years  ?  \        \ 

QUESTIONS.  \J^ 

183.  What  is  a  denomination  1  184.  A  denominate  number  1 
185.   A  simple  number  ?     186.    A  compound  number  ? 

187.  What  is  reduction  descending  ?  188.  How  is  a  denominate 
number  reduced  to  smaller  denominations  ? 

189.  What  is  reduction  ascending?  190.  How  is  a  denominate 
number  reduced  to  larger  denominations  ? 

191.  What  is  a  denominate  fraction  ?  192.  How  is  a  denominate 
fraction  reduced  to  integers  of  smaller  denominations  ?  1 93.  How 
are  denominate  integers  reduced  to  fractions  of  larger  denominations  ? 

196.  How  is  the  difference  between  dates,  in  yeais,  months,  and 
days,  found  ? 


THE    METKIC    SYSTEM.       /  141 


THE   METRIC    SYSTEM. 

197.  The  Metric  System  of  weights  and  measures,  now 
coming  into  use  in  the  United  States,  has  for  its  base  a  unit 
called  the  meter. 

Note.  —  This  system,  in  extensive  use  in  the  arts  and  sciences,  adopted  for 
the  United  States  Coast  Survey,  and  partially  employed  in  the  Mint  and  Gen- 
eral Post  Office,  was  legalized  for  use  in  the  United  States  by  Congress  in  1866. 

198.  The  Meter,  which  was  intended  to  be,  and  is  very 
nearly,  the  ten-millionth  part  of  the  distance  on  a  meridian 
from  the  equator  to  the  2)ole,  is  the  principal  unit  of  lengths, 
and  the  standard  unit  from  which  all  metric  measures  are 
derived. 

199.  The  Are,*  the  principal  unit  of  the  measures  of 
land,  is  a  square  whose  side  is  ten  meters. 

200.  The  Stere,  the  principal  unit  of  the  measures  of 
wood  and  stone,  is  a  cube  whose  edge  is  a  meter. 

201.  The  Liter,  the  principal  unit  of  the  measures  of 
capacity,  is  a  cube  whose  edge  is  the  tenth  of  a  meter. 

202.  The  Gram,  the  principal  unit  of  weight,  is  the 
weight  of  a  cube  of  pure  water  at  its  greatest  density, 
whose  edge  is  a  hundredth  part  of  a  meter. 

203.  The  Names  of  the  divisions  of  the  unit  are  formed 
by  prefixing  to  the  name  of  the  unit  the  Latin  words,  milli 
for  1000th,  ce7iti  for  100th,  and  deci  for  10th;  and  the 
names  of  multiples,  by  prefixing  the  Greek,  deka  for  10, 
hekto  for  100,  kilo  for  1000,  and  myria  for  10000. 

*  Are  is  pronounced  air ;  stere,  stair ;  and  liter,  lee'ter.  All  metric  names 
have  the  accent  on  the  first  syllable. 


142 


THE    METRIC    SYSTEM. 


204.  In  the  metric  system,  as  in  United  States  money, 
only  a  few  of  the  denominations  are  much  used.  These 
will  be  distinguished  in  the  tables  by  the  difference  in  type. 

The  unit  corresponds  to  the  dollar,  and  dcci^  centi,  milli 
to  climes,  cents,  mills. 


205.    LENGTH  MEASURES. 


10  millimeters  (^^)  are  1  centimeter, « 


10  centimeters 
10  decimeters 
10  meters 
10  dekameters 
10  hektometers 
10  kilometers 


1  decimeter,  ^™. 

1  METER,-  '". 

1  dekameter,  ^"^. 
1  hektometer,  ""^. 
1  kilometer,^'". 

1  myriameter,  ^^"^. 


Equivalents. 

1  centimeter  =  0.3937  inch. 
1  decimeter  =  3.937  inches. 
1  meter  =  39.37  inches. 

1  kilometer    =    0.6214  mile. 

1.  The  meter  is  used  in  measuring 
woven  fabrics  and  short  lengths  and  dis- 
tances. 

2.  The  kilometer  is  the  unit  in  meas- 
uring roads  and  long  distances. 

3.  The  decimeter  is  nearly  4  inches  ; 
the  m^ter,  about  3  feet  3|  inches  ;  and  the 
kilometer,  about  200  rods,  or  |  of  a  mile. 

206.  A  Metric  Number  is  writ- 
ten with  the  decimal  point  separat- 
ing the  unit  from  its  decimal  parts. 
Thus, 

;i^Km  gHm  yDm  3  m  ^dm  2  cm  ^^ittcu  as  mctcrs,  is  1573.42' 
and,  written  as  kilometers,  is  1.57342  ^'". 


^l 

. 

•"'      ~: 

tD        — 

E 

luches. 
1                                          2 

ill           ill           I     1      1     1     1      1 

CO         — 

UN    INI    II  1  1  Ml  II 

4                5 

Centimeters. 

OJ        - 

"*      - 

CO 

QO           — 

o        - 

THE    METRIC    SYSTEM.  143 

207.  In  reading  metric  numbers,  the  name  of  the  unit 
may  be  applied  to  all  on  the  left  of  the  decimal  point,  and 
the  name  of  the  smallest  denomination  denoted  to  all  on 
the  right  of  the  point.     Thus, 

42.73  ™  may  be  read  forty-two,  and  seventy-three  hun- 
dredths meters;  or,  forty-two  meters,  and  seventy-three 
centimeters. 

8.675  ^"^  may  be  read  eight,  and  six  hundred  seventy-five 
thousandths  kilometers ;  or,  eight  kilometers,  and  six  hun- 
dred seventy-five  meters. 

1.  How  many  meters  and  hundredths  of  a  meter  are  ex- 
pressed by  7.25  •"  ? 

2.  How  many  kilometers  and  thousandths  of  a  kilometer 
are  expressed  by  8.407  ^"^  ? 

3.  How  many  meters  and  centimeters  does  7.25  "  express  ? 

4.  How  many  kilometers  and  meters  does  8.407  ^™  ex- 
press ? 

5.  How  many  centimeters  in  4.15 '"  ?  How  many  meters 
in  7.384  ^^"  ? 

6.  Reduce  784  centimeters  to  meters  ;  6453  meters  to  kilo- 
meters. 

208.    SURFACE  MEASURES. 

100  square  millimeters  (^^i  ™™)  are  1  square  centimeter,  ^^  ^"^. 
100  square  centimeters  **   1  square  decimeter,  sqdm^ 

100  square  decimeters  "   1  square  meter,  s^'",  or  centare. 

100  square  meters  **    1  square  dekameter,  sqDm^ 

100  square  dekameters  "  1  square  hektometer,  sqHm,  • 

100  square  hektometers  "  1  square  kilometer,  sq^m^ 

Also, 

100  centares  («»),  or  sq.  meters,  are  1  are,  a. 

100  ares  ••   1  hektare,  "*. 


144  THE   METRIC    SYSTEM. 

Equivalents. 

1  square  centimeter  =  0. 155  0  sq.  inch.         1  are  =  3. 954  sq.  rods. 

1  square  decimeter   =  0. 1076  sq.  foot.         1  hektare         =  2. 471  acres. 
1  square  meter  =  1.196  sq.  yards.        1  sq. kilometer =0.38 61  sq.  mile. 

1.  The  square  meter  is  used  in  measuring  ordinary  surfaces  ;  the 
square  kilometery  in  measuring  the  area  of  countries  ;  and  the  a/re  and 
hektare^  in  measuring  land. 

2.  The  square  meter  is  about  lOf  square  feet,  or  1-|-  square  yards ; 
and  the  hektare,  about  2-|^  acres. 

3.  As  100  units  of  a  smaller  denomination  make  a  unit 
of  a  denomination  next  larger,  the  scale  is  100,  and  two 
places  of  figures  must  be  allowed  for  each  denomination. 

Thus,  ^1  Square 

'  Centimeter. 

31  "^  14  ^  17  *=%  written  as  ares,  is  3114.17  %  which  may 
be  read  3114  ares,  and   17  centares;   and,  written   as   hektares,  is 
31.1417  "%  which  may  be  read  31  hektares,  and  1417  centares. 

7.  In  4  square  meters  how  many  square  decimeters  ?  In  4 
square  centimeters  how  many  square  millimeters  ? 

8.  Express  65.41  ^  as  centares  ;  as  hektares. 

9.  How  many  ares  in  5734  ""^  ?  How  many  hektares  in 
6893  ^  ? 

209.    VOLUME  MEASURES. 

1000  cubic  millimeters  (cumm)  ^pe  1  cubic  centimeter,  <="«=™. 
1000  cubic  centimeters  "   1  cubic  decimeter,  cu^m^ 

1000  cubic  decimeters  **  1  cubic  meter,  '="™,  or  stere. 

Also, 

10  decisteres  (^^^)  are  1  stere,  ^t. 

^  Equivalents. 

1  cubic  centimeter  =    0.061  cu.  inch.      1  cubic  meter  =  1.308  cu.  yards, 
1  cubic  decimeter   =  61.022  cu.  inches.  1  stere  =  0.2759  cord. 

1.  The  cvhic  meter,  the  unit  of  ordinary  solids,  takes  the  name  of 
stere  when  applied  to  the  measuring  of  wood  and  lumber. 


THE   METRIC    SYSTEM. 


145 


1  Cubic 
Centimeter. 


2.  The  cubic  decimeter  is  about  61  cubic  inches  ;  the  cubic  meter,  or 
stere,  about  35j  cubic  feet. 

3.  Where,  as  in  the  table,  1000  units  of  a  smaller 
denomination  make  a  unit  of  a  denomination  next 
larger,  the  scale  is  1000,  and  three  places  of  figures 
must  be  allowed  for  each  denomination.     Thus, 

13cum  4ogcudm  573  cu  cm^  Written  as  cubic  meters, 
is  13.406578  '^^  ™  which  may  be  read  13  cubic  meters, 
and  406578  cubic  centimeters  ;  and,  written  as  cubic  decimeters,  is 
13406.578  ^"^'^ 

10.  In  6  "^""^  how  many  cubic  decimeters  ?     In  8  ''"^"'  how 
many  cubic  centimeters  ? 

11.  In  7000  '^^  '"'^  how  many  cubic  centimeters  ?     In  9000 
cu  dm  j^Q^  many  cubic  meters  ? 

12.  Express    76.006^"'^"'    as   cubic   centimeters;    as   cubic 
meters. 

13.  Express  0.3125  ''""'  as  cubic  decimeters  ;  as  cubic  centi- 
meters. 

210.    CAPACITY  MEASUKES. 

10  milliliters  Q^^)  are  1  centiliter,  ^^ 
10  centiliters  "    1  deciliter,  ^^ 

10  deciliters  "    1  liter,  K 

10  liters  "    1  dekaliter,  ^K 

10  dekaliters  "    1  hektoliter,  "i. 

10  hektoliters  "    1  kiloliter,  ^K 

Equivalents. 

1  liter  =  61.022  cu.  inches.  1  hektoliter  =    3.531  cu.  feet. 

1  liter  =    1.0567  liquid  quarts.  1  hektoliter  =  26.417  gallons. 

I  liter  =    0.908  dry  quart.  1  hektoliter  =    2.837  bushels. 


1  MiUiliter  = 


10 


- 


1  Cubic  Centimeter. 


146 


THE   METEIC    SYSTEM. 


1.  The  liter  is  used  in  measuring  liq- 
uids  ;  and  the  hektoliter  is  used  in 
measuring  grains,  roots,  and  liquids  in 
casks. 

2.  The  liter  is  about  1.06  liquid 
quarts,  or  0.9  of  a  dry  quart ;  and  the 
hektoliter  is  about  26^  gallons,  or  2| 
bushels. 


14.  How  many  liters,  and  what  decimal  of  a  liter,  are  ex 
pressed  by  6.45  *  ? 

15.  How  many  kiloliters,  and  what  decimal  of  a  kiloliter, 
are  expressed  by  9.750  ^^  ? 

16.  How  many  centiliters  are  6  liters  ?     7.55  liters  ? 

17.  How  many  liters  in  600  centiliters  ?     How  many  kilo 
liters  in  6000  liters  ? 


211.    WEIGHT  MEASURES. 


10  milligrams  (""s^)  are  1  centigram,  <=ff. 

10  centigrams  "  1  decigram,  ^s, 

10  decigrams  "  1  gram,  st. 

10  grams  "  1  dekagram,  ^s, 

10  dekagrams  "  1  hektogram,  "«, 

10  hektograms  "  1  kilogram,  ^^,  or  ^^ 

10  kilograms  "  1  myriagram,  ^s, 

10  myriagrams  "  1  quintal,  Q. 

10  quintals  "  1  metric  ton,  '^. 


THE  METRIC   SYSTEM.  147 

Equivalents. 

In  weight 

1  gram  =  1  cu.  centimeter,  or  1  milliliter  of  water 

1  kilogram    =  1  cu.  decimeter,  or  1  liter  of  water. 
1  metric  ton  =  1  cu.  meter,  or  1  kiloliter  of  water. 

1  gram  =  15.432  grains  troy.  1  kilogram    =:  2.2046  pounds  av 

1  gram  =    0.03527  ounce  av.  1  metric  ton==  1.1023  tons. 

Note.  —  The  weight  of  the  gram  is  determined  when  the  water  is  pure  and 
at  the  temperature  of  its  greatest  density,  which  is  39. 2°  Fahrenheit. 

1.  The  gram  is  used  in  weighing  letters,  gold,  and  jewels,  and  in 
mixing  medicines  ;  the  hilogram,  or,  for  brevity,  kilo^  is  used  in 
weighing  common  articles  ;  and  the  metric  ton,  in  weighing  very 
heavy  articles. 

2.  The  kilo  is  about  ^\  pounds  ;  and  the  metric  ton,  about  1^ 
common  tons. 

3.  Of  the  United  States  coinage,  the  nickel  five-cent  piece  weighs 
5  grams  ;  two  silver  half-dollars,  25  grams  ;  and  80  silver  half-dollars, 
a  kilo.  • 

4.  A  letter  sent  for  a  single  postage  must  not  exceed  the  weight  of 
six  nickels,  or  30  grams. 

18.  In  5.65  ^,  how  many  decigrams  ?  How  many  centi- 
grams ? 

19.  In  5650  ""^j  how  many  centigrams  ?  How  many  deci- 
grams ? 

20.  In  6.315  ^^,  how  many  grams  ?  In  4.670  "^^  how  many 
kilos  ? 

21.  What  decimal  part  of  a  kilo  is  a  gram  ?  What  decimal 
part  of  a  metric  ton  is  a  kilo  ? 

22.  At  20  cents  a  kilo,  how  many  dollars  will  a  metric  ton 
of  sugar  cost  ? 

23.  A  tank  has  the  capacity  of  5.250  kiloliters.  How  many 
metric  tons  of  pure  water  will  it  hold  ? 

24.  I  wish  to  weigh  8.75  ^^.  How  many  silver  half-dollars 
will  serve  for  the  weights  ? 


148  tHE  METRIC   SYSTEM. 


REDUCTION  OF  UNITS. 

212.  The  Units  shown  in  the  metric  tables  form  a  deci- 
mal system  (Art.  19),  to  which  apply  the  folio  whig 

Principles. 

1.  Ten  units,  or  some  multiple  of  ten  units,  of  any  de 
nomhiation  make  one  of  the  next  larger  unit. 

2.  A  metric  number  may  he  changed  from  one  denomination 
to  another  next  smaller,  or  larger,  by  moving  the  decimal 
point  one  or  more  places  to  the  right,  or  left,  as  the  case  may 
he, 

3.  Any  denomination  may  he  taken  as  the  unit,  the  num- 
ber at  the  right  of  the  point  being  read  as  a  decimal  of  the 
unit. 

213.  The  units  of  the  metric  and  the  common  system 
may  be  readily  compared  by  means  of  the  equivalents 
which  have  been  given,  and  by  means  of  the  following 

COMPARATIVE    TABLE. 

Length.  1  cu.  yard  =    0.7646  cu.  meter. 

1  inch   =    2.54  centimeters.  1  ^'^^^         =    3.625  steres. 

1  foot    =  30.48  centimeters. 

1  rod     =    5.029  meters.  Capacity. 

1  mile   =    1.6093  kilometers.  1  liq.  qnart  =    0.9465  liter. 

«     -  1  gallon        =    3.785  liters. 

1  dry  quart  =    1.101  liters. 
1  sq.  inch   =    6.452  sq.  centim.         ^  ^^^^^        ^    0  3^24  hektoliters. 
1  sq.  foot    =    9.2903  sq.  decim. 
1  sq.  yard  =    0.8361  sq.  meter. 
1  sq.  rod     =    0.2529  are. 
1  acre  =    0.4047  hektare.  1  grain  troy    =    0.648  centigram. 

1  sq.  mile   =    2.59  sq.  kilometers.     1  ounce  troy  =  31.1035  grams. 

Volwme.  1  o^^nco  av.     =  23.35  grams. 

1  cu.  inch  =  16.387  cu.  centimeters.  1  j)oiind  av.     =    0.4536  kilogram. 
1  cu.  foot  =  28.317  cu.  decimeters.     1  comnsonT.  =    0.9072  mot.  ton. 


Weight, 


THE  METRIC  SYSTEM.  149 

WRITTEN    EXERCISES. 

25.  Express  as  meters  and  add  1365  ™",  497  ^"'j  and  145.51  '"c 

26.  Express  as  ares  and  add  15.16  "^  111.55  %  and  3615  '=\ 

27.  Erom  a  range  of  wood  containing  45  steres,  I  have  sold 
276  decisteres.     How  many  steres  remain  ? 

28.  How  many  liters  in  6  casks^  each  containing  3.40  ^  ? 

29.  Eight  men  shared  equally  21.080  metric  tons  of  sugar. 
What  is  each  man's  share  worth  at  20  cents  a  kilo  ? 

30.  Erom  a  farm  containing  365.50  "^  there  have  heen  sold 
two  small  lots ;  the  one  containing  8.42  ^^,  and  the  other 
87.25  ^.     How  much  remains  ? 

31.  Change  125  meters  to  feet. 

39.37  in.  Solution.  —  As  1  meter  =  39.37 

1^5  inches,  125  meters  must  equal  125 

19685  times    39.37     inches,   or    4921.25 

7874  inches.      As   1  foot  =  12  inches, 

3937  there  will  be  as  many  feet  as  12 

12  in.)  4921  25  in.  inches    are     contained    times    in 

410  IOt^    ft  4921.25  inches,  or  410.10i5^  ft. 

32.  If  your  weight  is  55  kilos,  what  is  it  in  pounds  avoir- 
dupois ? 

33.  A  garden  plat  contains  306  square  meters.  How  many 
square  yards  are  there  in  it  ? 

34.  A  farm  is  450  hektares  in  extent.  How  many  acres 
does  it  contain  ? 

35.  A  barrel  of  flour  weighs  196  pounds.  What  is  its 
weight  in  kilos  ? 

36.  The  produce  of  7  acres  was  210  bushels  of  wheat.  What 
was  it  in  hektoliters  ? 

37.  When  butter  is  35  cents  a  pound,  how  much  should  it 
be  a  kilo  ? 

Solution.  — At  35  cents  a  pound,  a  kilo,  which  is  2.2046  pounds, 
must  cost  2.2046  times  35  cents,  or  77+  cents. 


150  THE  METRIC  SYSTEM. 

38.  At  65  cents  a  bushel,  what  should  be  the  price  of  corn 
a  hektoliter  ? 

39.  What  is  the  value  of  an  eighth  of  a  plantation  of  600.58 
hektares  at  $  25  an  acre  ? 

40.  The  distance  by  railroad  between  Boston  and  New 
Orleans  is  1607  miles.     What  is  it  in  kilometers  ? 

►  41.  The  dome  of  the  capitol  at  Washington  is  287  ft.  6  in 
high,  surmounted  by  a  statue  of  Liberty  19  ft.  6  in.  high. 
What  is  the  whole  height  in  meters  ? 

42.  Bought  a  roll  of  carpeting  of  65  yards  at  1 1.20  a  meter, 
and  sold  it  at  the  same  price  a  yard.  How  much  did  I  make 
by  the  transaction  ? 

43.  The  capacity  of  a  certain  bin  is  40.64  cubic  meters. 
What  is  the  value  of  the  grain  that  can  be  put  in  it  at  80  cents 
a  bushel  ? 

QUESTIONS. 

197.  What  is  the  metric  system?     198.  What  is  a  meter  ?     199. 
An  are  ?     200.  A  stere  ?    201.  A  liter  ?    202.  A  gram  ? 
203.  How  are  the  names  of  the  divisions  of  the  metric  units  formed  1 

205.  Recite  the  table  of  metric  measures  of  length.  For  what  is 
the  meter  used  1    The  kilometer  ? 

206.  How  is  a  metric  number  written  1  207.  How  are  metric 
numbers  read  ? 

.  208.  Recite  the  table  of  metric  measures  of  surface.  How  is  the 
square  meter  used  ?     The  square  kilometer  ?     The  are  and  hektare  1 

209.  Name  the  measures  of  volume.  How  is  the  cubic  meter 
used  ?    When  does  it  take  the  name  of  the  stere  ? 

210.  Recite  the  table  of  measures  of  capacity.  How  is  the  liter 
used  ?     The  hektoliter  1 

211.  Recite  the  table  of  measures  of  weight.  For  what  is  the  giam 
used  ?    The  kilogram  ?     The  metric  ton  1 

212.  How  many  units  of  one  denomination  make  a  unit  of  another 
in  the  metric  system  ?  How  may  a  metric  number  l)e  changed  from 
one  denomination  to  another  ?  When  any  denomination  is  taken  as 
the  unit,  how  may  the  number  at  the  right  of  the  point  be  read  ? 


MEASUKEMENTS.  151 

MEASUREMENTS. 

SURFACES. 

214i  1.  How  many  square  inches  in  a  surface  8  inches 
long  and  1  inch  wide  ?  In  a  surface  8  inches  long  and  2  inches 
wide  ? 

2.  A  path  12  feet  long  and  2  feet  wide  has  how  many  square 
feet  of  surface  in  it  ? 

3.  A  table  is  6  feet  long  and  4  feet  wide.  How  many  square 
feet  of  surface  has  it  ? 

215.  A  Plane  Figure  is  a  portion  of  a  plane  surface 
(Art.  163)  bounded  by  lines. 

216.  The  Perimeter  of  a  plane  figure  is  the  sum  of  its 

bounding  lines. 

217.  The  Area  of  a  plane  figure  is  the  surface  included 
within  its  perimeter. 

218.  The  Dimensions  of  a  rectangle 
are  its  length  and  breadth. 

A  rectangle  3  inches  long  and  2    ^^H^^HK 
inches  broad  contains  in  one  row  3  " 

squares  of  1  square  inch  each ;  and  2  such  rows  contain 
2  times  3  square  inches,  or  6  square  inches.     That  is, 

The  area  of  a  rectangle  is  equal  to  the  product  of  its  length 
and  breadth,  taken  in  the  same  denomination. 
Also, 

One  of  the  dimensions  of  a  rectangle  is  equal  to  the  area 
divided  hy  the  other  dimension. 


152 


MEASUREMENTS. 


219.  A  Triangle  is  a  plane  figure 
bounded  by  three  straight  lines. 

The  Base  of  a  triangle  is  the  line 
upon  which  it  stands  ;   and  the  Alti- 
tude is  its  height  above  the  base,  or 
^    the  base  extended.     Thus, 

^  ^  is  the  base,  and  C  B  the  altitude,  of  the  triangle 
A  BG, 

220.  Draw  the  lines  A  E,  D  B  per- 
pendicular to  the  extremities  of  the 
base  of  the  triangle  A  C  B,  and  draw 
the  line  E  D  through  C,  parallel  to 
A  B,  and  it  is  evident  that  the  triangle 
A  0  Bis  half  the  rectangle  ABBE, 

of  the  same  base  and  altitude.     That  is. 

The  area  of  a  triangle  is  half  the  area  of  a  rectangle  of 
the  same  base  and  altitude. 


221.  A  Circle  may  be  regarded  as 
consisting  of  a  great  number  of  tri- 
angles, whose  bases  form  the  circum- 
ference of  a  circle,  and  whose  altitude 
is  the  radius  of  the  circle.     Hence, 


The  area  of  a  circle  is  equal  to  half  the  product  of  the  cir- 
cumference hy  the  radius, 

222.  The  quotient  of  the  circumference  of  a  circle  di- 
vided by  the  diameter,  to  the  nearest  ten-thousandth,  is 
3.1416.     Hence, 

The  circumference  is  equal  to  the  diameter  multiplied  hy 
3.1416  ;  tlu  diameter  is  equal  to  the  circumference  divided 
by  3.1416. 


62.75 

26 

31660 

10660 

1371.60  sq. 

ft. 

1371  sq.  ft. 

72 

sq. 

in. 

MEASUKEMENTS.  153 


WRITTEN    EXERCISES. 

4.   What  is  the  area  of  a  floor  which  is  52  feet  9  inches  long 
and  26  feet  wide  ? 

Solution.  ^52  ft.  9  in.  =  52.V5 
feet.  The  product  of  the  length  by 
the  width  gives  as  the  number  of 
square  feet  of  surface  1371.50  sq.  ft., 
or  1371  sq.  ft.  72  sq.  in.,  which  is  the 
Or,  area  required. 


5.  The  area  of  a  floor  is  1371  square  feet  72  square  inches, 
and  its  width  is  26  feet.     What  is  its  length  ? 

52.75  ft.  =  52  ft.  9  in. 

26)  1371.50  Solution.  —  1371  sq.  ft.  72  sq. 

130  in.  =  1371.50 sq.ft.   Asthearea 

71  1371.50  sq.  ft.  is  the  product  of 

g2  ^he  length  and  width,  the  width 

YqF  must  equal  the  quotient  of  the 

^^^  area    1371.50    divided  by   the 

i^  length  26,  which  is  52.75  ft.  = 

^^^  52  ft.  9  in. 

130 

6.  The  base  of  a  triangle  is  46  ft.  3  in.,  and  the  altitude 
35  ft.  6  in.     What  is  its  area  ? 

7.  The  circumference  of  a  circle  is  314.16  feet,  and.  its  ra= 
dius  50  feet.     What  is  its  area  ? 

8.  The  diameter  of  a  circle  is  400  feet.     What  is  its  cir- 
cumference ? 

9.  The  circumference  of  a  circle  is  1256.64  feet.     What  is 
its  diameter  ? 

10.  How  many  yards  of  carpeting  1  yard  wide  will  be  re- 
quired to  carpet  a  room  18  ft.  long  and  15  ft.  6  in.  wide  ? 


154  MEASUREMENTS. 

11.   A  room  that  is  18  ft.  9  in.  square  requires  50  yards  of 
carpeting.     What  is  the  width  of  the  carpeting  ? 
*r*  12.   How  many  acres  in  a  field  45  rods  long  and  48  rods 
wide  ? 

13.  The  diameter  of  a  circle  is  15  meters.  What  is  its  cir- 
cumference in  feet  ? 

14.  What  is  the  area  of  the  gable  end  of  a  house  32  ft 
wide,  the  ridge  being  14  ft.  6  in.  higher  than  the  base  of  the 
gable  ? 

.  15.   How  many  hektares  in  a  rectangular  meadow  564.50 
meters  long  and  260  meters  wide  ? 

16.  What  will  it  cost  at  60  cents  a  square  yard  to  concrete 
a  walk  288  ft.  long  and  12  ft.  6  in.  wide  ? 

17.  A  horse  is  fastened  to  a  stake  by  a  chain  60  feet  long. 
How  many  square  rods  of  surface  can  the  chain  sweep  over  ? 

18.  How  many  square  feet  of  sheet  zinc  will  be  required  to 
line  a  cistern  6  ft.  deep,  having  a  square  bottom,  of  which 
each  side  is  2  ft.  6  in.  ? 

19.  The  capitol  at  Washington  is  751  feet  long  and  348  feet 
wide.     How  many  acres  does  it  cover  ? 

20.  On  laying  the  pavement  of  a  court  with  stones  2  ft.  6 
in.  long  by  9  in.  wide,  it  is  found  that  it  requires  75  stones  to 
form  one  strip  extending  the  whole  length  of  the  court,  and 
that  8 J  strips  will  exactly  cover  it.  What  is  the  area  in 
square  yards,  and  what  is  the  cost  of  the  pavement  at  20 
cents  a  square  foot  ? 

VOLUMES.  ^ 

21.  How  many  cubic  feet  in  a  beam  10  feet  long,  1  foot  wide, 
and  1  foot  deep  ?  In  a  beam  10  feet  long,  1  foot  wide,  and  2 
feet  deep  ? 

22.  How  many  cubic  feet  in  a  block  of  marble  8  feet  long, 
2  feet  wide,,  and  1  foot  thick  ?  In  a  block  8  feet  long,  2  feet 
wide,  and  2  feet  thick  ? 


MEASUREMENTS. 


155 


223.  A  Rectangular  Volume   is  a 

body  bounded  by  six  rectangles. 

The    Dimensions   of   a   rectangular 
volume  are  its  length,  breadth,  and 

tniCKneSS.  a  RectaDguliar  Volume. 

224.  The  Contents  of  a  rectangular  volume  are  the  space 
contained  within  its  bounding  surfaces. 

The  dimensions  of  a  rectangular  volume  determine  its 
contents.     Thus, 

A  rectangular  volume  3  inches  long,  3 
inches  wide,  and  2  inches  thick,  contains 
in  one  layer  3  rows  of  3  viubic  inches,  or  9 
cubic  inches,  and  2  such  layers,  or  2  times 
3  times  3  cubic  inches,  contain  18  cubic 
inches.     That  is. 

The  contents  of  a  rectangular  volume  are  equal  to  the  pro- 
duct of  its  length,  breadth,  and  thickness^  taken  in  the  same 
denomination. 
Also, 

One  of  the  dimensions  of  a  rectangular  solid  equals  its 
contents  divided  hy  the  product  of  the  other  two  dimensions. 

225.  A  Cylinder  is  a  round  body  of  uni- 
form diameter,  whose  bases  are  equal  and 
parallel  circles. 

The  altitude  of  a  cylinder  is  the  straight 
line  joining  the  centers  of  the  two  bases  ;  as 
the  line  A  B. 

226.  The  Contents  of  a  cylinder  are  equal 

to  the  product  of  the  area  of  the  base  by  the  altitude. 

The  Cu7^ed  Surface  of  a  cylinder  is  equal  to  the  ^product 
of  its  circumference  and  altitude. 


^ 


156  MEASUREMENTS. 


WRITTEN     EXERCISES. 


23.  How  many  cubic  feet  in  a  rectangular  block  8  feet  long, 
4  feet  6  inches  wide,  and  2  feet  3  inches  thick  ? 

Solution.  —  4  ft.  6  in.  =  |  ft. ;  2  ft.  3  in.  =  |  ft.  The  product  of 
the  three  dimensions,  8  X  f  X  |  =  81  cu.  ft. 

24.  The  contents  of  a  rectangular  block  8  feet  long  and  4 J 
feet  wide  are  81  cubic  feet.     What  is  its  thickness  ? 

8  X  4i  :=  36  Solution.  —  The  contents  of  the  block 

are  the  product  of  its  three  dimensions. 

81  -^  36  =  24-  The  product  of  the  length  and  width  is 

36.     The   given   product  of  the  three 

24  ft.  =  2  ft.  3  in.       dimensions  divided  by  36  gives  for  the 

thickness  2^  ft.,  or  2  ft.  3  in. 

25.  What  are  the  contents  of  a  trunk  4  feet  long,  2^  feet 
wide,  and  18  inches  deep  ? 

26.  How  many  cubic  feet  of  space  in  a  room  whose  dimen- 
sions are  each  10  feet  6  inches  ? 

27.  A  rectangular  cistern,  whose  length  is  13|  feet  and 
breadth  6  feet,  contains  294J  cubic  feet  of  water.  What  is 
the  depth  of  the  water  ? 

28.  A  box  is  2  meters  long,  15  decimeters  wide,  and  1  meter 
deep.     What  is  its  capacity  ? 

29.  How  many  cubic  yards  of  earth  must  be  removed  in  ex- 
cavating a  cellar  60  feet  long,  42  feet  wide,  and  8  feet  6  inches 
deep  ? 

30.  A  block  is  3  meters  long,  2  meters  wide,  and  1.45  me- 
ters thick.     What  are  its  contents  ? 

31.  A  circular  well  is  32  feet  deep  and  3  feet  in  diameter. 
How  many  cubic  yards  does  it  contain  ? 

32.  A  roller  is  4  feet  6  inches  long,  and  6  feet  4  inches  in 
circumference.  How  much  surface  will  it  pass  over  in  revolv- 
ing 36  times  ? 


MEASUREMENTS. 


157 


WOOD    MEASURE. 


-/?  ,. .-.       A  COKD 


227.  A  Range  of  wood  8  feet  long,  4  feet  wide,  and  4 
feet  high,  makes  a  cord  of  wood  (Art.  171). 

A  Cord  Foot  is  1  foot  in  length  of  this  range,  or  16  cubic 
feet. 

WRITTEN    EXERCISES. 

33.  How  many  cords  of  wood  in  a  range  32  feet  long,  8  feet 
high,  and  4  feet  wide  ? 

34.  How  many  cords  of  4-foot  wood  in  a  range  64  feet  long 
and  4  feet  high  ? 

35.  A  range  of  4-foot  wood  is  28  feet  long  and  6^  feet  high. 
How  many  cords  does  it  contain  ? 

36.  A  range  of  4-foot  wood  is  56  feet  long.  How  high 
must  it  be  piled  to  contain  10  cords  ? 

37.  Wood  is  loaded  upon  a  cart  in  2  piles,  3  feet  6  inches, 
wide  and  4  feet  6  inches  high.  Allowing  the  wood  to  have 
been  cut  4  feet  long,  how  much  wood  is  there  on  the  cart  ? 

38.  A  shed  contains  a  pile  of  wood  30  feet  long,  16  feet  wide, 
and  12  feet  high.  What  is  the  value  of  the  wood  at  $  5.25  a 
cord  ? 


158  MEASUREMENTS. 


BOARD    MEASURE. 

228.  A  Board  Foot  is  a  square  foot  of  board  one  inch 
thick;  and 

12  board  feet  make  1  cubic  foot. 

229.  Lumber,  or  sawed  timber,  as  boards,  plank,  joists, 
and  the  like,  is  estimated  in  board  feet. 

Heivn  Timber  is  estimated  either  in  board  feet  or  cubic 
feet. 

In  the  measurement  of  lumber,  or  hewn  timber  which 
tapers,  half  the  width  or  half  the  thickness  of  the  two  ends 
is  taken. 

230.  To  find  the  contents  of  a  piece  of  lumber  in  board 
feet, 

Multiply  the  product  of  the  length  and  width  taken  in 
feet  by  the  thickness  in  inches. 

Note.  —  Lumber  less  than  an  inch  thick  is  measured  as  if  it  were  an  inch 
thick.  In  computations,  therefore,  thickness  may  be  disregarded  if  an  inch  or 
less. 

In  the  measurement  of  box  boards,  however,  the  standard  of  thickness  is  § 
of  an  inch. 

WRITTEN    EXERCISES. 

39.  How  many  board  feet  in  a  joist  21  feet  6  inches  long, 
4  inches  wide,  and  3  inches  thick  ? 

Solution.  —  Multiplying   to- 
21  ft.  6  in.  =  -^2^-  ft.  gether    the    length  and  width 

expressed  in  feet,  we  have,  as 
4  inches    =  i%  =  g-  ft.  surface  feet,  ^  x  ^,  and,  mul- 

tiplying by  the  inches  in  thick- 
"V"  X  i  X  ^  =  ^^i-  ^^*  ^^'        ne^s,  have  as  board  feet,  ^  x  J 

X  3  =  21^. 

40.  How  many  board  feet  in  a  board  16  feet  long,  1  foot 
6  inches  wide,  and  |  inch  thick  ? 


MEASUREMENTS.  159 

41.  Required  the  contents  in  board  feet  of  a  plank  20  feet 
long,  16  inches  wide,  and  2 J  inches  thick? 

42.  How  many  board  feet  in  a  piece  of  hewn  timber  20  feet 
long,  10  inches  thick,  and  whose  width  tapers  from  18  to  16 
inches  ? 

43.  What  will  be  the  cost  of  44  spruce  joists,  each  18  feet 
long,  9  inches  wide,  and  3  inches  thick,  at  $  23  a  thousand 
feet  ? 

44.  What  is  the  value  of  18  planks  2  inches  thick,  24  feet 
long,  21  inches  wide,  at  $  35  a  thousand  feet  ? 

MISCELLANEOUS    EXERCISES. 

45.  A  rectangular  lot  containing  9432  square  feet  has  a 
frontage  of  131  feet.     What  is  its  other  dimension  ? 

46.  A  wagon  8  feet  long  and  3  feet  wide  has  wood  piled  on 
it  5^-  feet  high.  How  much  is  the  wood  worth  at  $  5.50  a 
cord  ? 

47.  How  many  board  feet  of  plank  2  inches  thick  will  be 
required  to  cover  a  platform  30  ft.  6  in.  long  and  20  ft.  wide  ? 

48.  What  must  be  paid,  at  the  rate  of  $  40  a  thousand  feet, 
for  6  boards  16  feet  long,  and  in  width  tapering  from  2S  to  20 
inches  ? 

49.  A  cistern  2.50  meters  long,  90  centimeters  wide,  and 
110  centimeters  deep,  will  hold  how  many  hektoliters  of  water  ? 

50.  How  many  rolls  of  paper  12  yards  long  and  1  foot  8 
inches  wide  will  be  required  to  paper  a  room  18  feet  long,  12 
feet  wide,  and  9  feet  high,  no  allowance  being  made  for  win- 
dows and  doors  ? 

51.  Allowing  .8  of  a  bushel  to  a  cubic  foot,  how  many 
bushels  of  corn  can  be  put  in  a  bin  whose  inside  measure  is 
4  feet  long,  3  feet  wide,  and  4  feet  deep  ? 

52.  Allowing  7J-  gallons  to  a  cubic  foot,  how  much  will  a 
tank  hold  whose  inside  measure  is  2  ft.  6  in.  long,  2  ft.  wide, 
and  1  ft.  9  in.  deep  ? 


160  MEASUREMENTS. 

53.  Allowing  for  each  square  foot  of  surface  of  a  brick  wall 
twice  as  many  bricks  in  number  as  the  wall  is  inches  thick, 
how  many  bricks  are  there  in  a  wall  40  feet  long,  12  feet 
6  inches  high,  and  8^  inches  thick  ? 

54.  If  a  ton  of  2000  pounds  of  Lehigh  coal  fills  40  cubic 
feet,  and  of  Lackawanna  coal  45  cubic  feet,  how  much  of  each 
kind  will  fill  a  bin  8  feet  long,  5  feet  wide,  and  4  feet  deep  ? 

55.  Water  is  flowing  into  a  cistern  whose  rectangular  base 
is  4840  square  inches.  How  many  cubic  feet  will  have  been 
supplied  when  the  depth  of  water  is  3J  feet,  and  what  w^ill  be 
its  w^eight  at  the  rate  of  1000  ounces  a  cubic  foot  ? 

56.  How  many  cubic  feet  in  the  walls  of  a  cellar,  whose 
length  is  33  feet,  width  30  feet,  and  depth  9  feet,  allowing  the 
thickness  of  the  walls  to  be  18  inches,  and  the  lap  of  the  one 
wall  by  the  other  at  each  corner  of  the  cellar  to  be  18  inches  ? 


-1- 


QUESTIONS. 


215.  What  is  a  plane  figure? 

216.  What  is  the  perimeter  of  a  plane  figure  ?  217.  The  area  of  a 
plane  figure  ?     218.  To  what  is  the  area  of  a  rectangle  equal  ? 

219.  What  is  a  triangle  ?  The  base  of  a  triangle  ?  The  altitude  ? 
220.  How  is  the  area  of  a  triangle  found  1 

221.  How  may  the  circle  be  regarded  ?  To  what  is  the  area  of  a 
circle  equal  ?  222.  To  what  is  the  circumference  equal  ?  To  what 
is  the  diameter  equal  ? 

223.  What  is  a  rectangular  volume  ?  What  are  the  dimensions  of  a 
rectangular  volume  ?  224.  The  contents  ?  225.  What  is  a  cylinder  ? 
The  altitude  of  a  cylinder  1  226.  What  are  the  contents  of  a  cylin- 
der ?     To  what  is  the  curved  surface  of  a  cylinder  equal  ? 

227.  What  range  of  wood  makes  a  cord  ?  How  much  of  such  a 
range  is  a  cord  foot  1 

228.  What  is  a  board  foot  1  229.  How  is  lumber  estimated  ?  How 
is  hewn  timber  estimated  1  230.  How  are  the  contents  of  a  piece  of 
lumber  found  ? 

224.  The  contents  of  a  room  divided  by  the  area  of  the  floor  will 
give  what  ? 


REVIEW.  161 


REVIEW. 

ORAL    EXERCISES. 

231.     1.   The  top  of  your  desk  is  2  ft.  long  and  18  in.  wide. 
How  many  yards  around  it  ? 

2.  It  is  IJ  miles  to  the  post-office.     How  many  miles  do  I 
travel  in  making  three  round  trips  to  it  ? 

3.  How  many  score  in  a  gross  ? 

4.  My  hens  lay  an  egg  apiece  daily.     How  many  dozen  do 
3  of  them  lay  in  3  weeks  3  days  ? 

5.  If  a  ton  of  coal  measures  40  cu.  ft.,  how  many  tons  will 
two  5-foot  cubical  bins  hold  ? 

6.  How  many  degrees  does  the  minute-hand  of  a  clock  move 
over  in  45  minutes  ? 

7.  What  is  the  perimeter  of  the  largest  surface  measure  ? 

8.  What  is  the  area  of  the  entire  surface  of  a  4-foot  cube  ? 

9.  How  wide  must  a  yard  of  ribbon  be  to  contain  144  sq.  in.  ?_/ 

10.  W^hat  will  it  cost  to  carpet  a  room  18  ft.  square  at  $  1  ' 
per  square  yard  ? 

11.  What  costs  a  mile  of  wire  at  a  cent  a  foot  ? 

12.  At  30  cents  a  gallon,  what  is  my  milk-bill  for  April  ?    I 
take  2  quarts  daily. 

13.  What  will  a  12-inch  cube  of  marble  cost  at  one  cent  per 
cubic  inch  ? 

14.  f  of  48  is  ^Y  of  what  number  ? 

15.  If  ^  ounce  of  tea  costs  5  cents,  what  will  2^  pounds 
cost  ? 

16.  How  many  acres  in  my  grandfather's  farm,  which  is  J 
of  a  mile  square  ? 

17.  If  3^-  dozen  cost  I  3.50,  what  will  y^^  <iozen  cost  ? 

18.  Bought  an  acre  of  land  at  $0.50  per  square  rod,  and 
sold  it  so  as  to  double  my  money.     Required  my  gain  ? 

19.  Bought  pencils  for  $1.25  a  gross,  and  sold  them  for  a 
cent  each.     How  much  did  I  gain  on  10  gross  ? 

11 


162  Heview. 

20.  How  long  must  a  field  be  that  is  20  rods  wide  to  contain 
3  acres  ? 

21.  What  will  7  lb.  15  oz.  of  sugar  cost  at  8  ^  per  pound  ? 

22.  A  school  contains  96  boys,  and  -^^  of  the  pupils  are  girls. 
How  many  pupils  in  the  school  ? 

23.  The  entire  surface  of  a  cube  is  96  square  inches.    What 
is  its  edge  ? 

24.  An  arc  of  80  degrees  is  what  fraction  of  a  semi-circum- 
ference ? 

25.  How  many  paper  boxes  3  in.  long,  2  in.  wide,  and  2  in. 
deep  can  be  packed  to  a  cubic  foot  ? 

26.  How  many  writing-books  of  24  pages  each  can  be  made 
from  \  a  ream  of  paper,  the  sheets  being  of  the  proper  size  ? 

27.  What  cost  18  dozen  blank-books  at  33J  cents  each  ? 

28.  I  sailed  for  Europe  May  17,  and  reached  home  again 
August  5.     How  long  was  I  away  ? 

29.  School  closes  July  10  for  a  6-weeks'  vacation.     When 
will  it  begin  again  ? 

30.  What  effect  has  moving  the  decimal  point  two  places 
to  the  left,  and  why  ? 

31.  What  is  the  circumference  of  a  circle  10  inches  in  di- 
ameter ? 

32.  How  many  4-inch  squares  in  a  square  foot  ? 

33.  How  many  days  in  the  five  shortest  months  of  a  leap 
year  ? 

34.  What  costs  a  stere  of  wood  at  I  0.75  per  decistere  ? 

35.  What  decimal  of  an  acre  is  a  plat  of  ground  15  rd.  long 
and  4  rd.  wide  ? 

36.  What  fraction  of  March  has  passed  at  noon,  March  25  ? 

37.  At  $  7  a  week,  what  is  my  board-bill  from  December  15 
to  January  22  ? 

38.  How  many  strips  of  carpeting  |  yd.  wide  will  be  re. 
quired  for  a  room  18  ft.  square  ?     How  many  yards  ? 

39.  Divide  0  48  by  0.2.     Find  their  product. 


REVIEW.  163 

40.  A  can  do  in  2  days  the  work  which  B  can  do  in  3  days. 
if  B  works  with  A,  in  how  many  days  will  the  work  be  done  ? 

41.  What  costs  a  strip  of  rosewood  4  ft.  long,  3  in.  wide, 
and  1^  in.  thick  at  60  cents  a  board-foot  ? 

42.  A  note  dated  August  11  is  payable  in  33  days.  On 
what  day  must  it  be  paid  ? 

43.  A  sidewalk  a  mile  long  contains  just  an  acre.  How 
wide  is  it  ? 

44.  If  J  of  a  pound  costs  $  J,  what  will  |  of  a  pound  cost  ? 

45.  I  give  yV  of  my  potatoes  to  Mr.  Murphy  for  digging 
them.  He  sells  his  share  for  $  8,  at  40  cents  a  bushel.  What 
can  I  get  for  my  share  at  the  same  rate  ? 

46.  Emma's  birthday  is  Aug.  16, 1872.  John  is  4  y.  8  mo. 
27  days  older.     When  is  his  birthday  ? 

47.  Name  the  natural  divisions  of  time. 

48.  A  cylinder  10  inches  long  has  f  of  a  square  inch  as  the 
area  of  its  base.  What  is  its  weight  if  each  cubic  inch  weighs 
2  ounces  ? 

49.  If  your  school-room  is  15  feet  high,  and  contains  12000 
cubic  feet,  how  many  square  feet  are  there  in  the  floor  ? 

WRITTEN    EXERCISES. 

50.  How  many  times  is  the  greatest  common  divisor  of  1116 
and  1364  contained  in  their  least  common  multiple  ? 

51.  Divide  the  product  of  0.008  and  0.06  by  0.16. 

52.  Change  yf  g  to  a  decimal. 

53.  Change  f  to  hundredths. 

54.  If  a  rail  is  33  feet  in  length,  how  many  rails  will  lay  a 
double  track  of  railroad  10  miles  long  ? 

55.  What  cost  17.65  tons  of  coal  if  $  2S  pays  for  3^  tons  ? 

56.  An  estate  is  valued  at  $  14000.  The  house  is  worth  f 
as  much  as  the  land.     Required  the  value  of  each. 

57.  I  have  a  square  mile  of  land.  I  sell  f  of  it  to  one  man, 
J  to  another,  and  100  A.  to  a  third.  How  many  acres  have  I 
left  ? 


164  EEVIEW. 

58.  I  have  $  570  in  my  safe,  which  is  $  80  less  than  f  of 
what  I  have  in  the  bank.     How  much  money  have  I  ? 

59.  The  runners  in  a  10-mile  race  go  how  many  times  around 
a  park  320  feet  long  and  300  feet  wide  ? 

60    Change  20000  square  inches  to  larger  denominations. 

61.  How  many  yards  of  carpeting  f  yd.  wide  will  be  re- 
quired for  a  parlor  28  ft.  by  15  ft.  ? 

62.  How  many  cubic  yards  of  earth  will  be  required  to  raise 
the  surface  of  an  acre  one  foot  ? 

^  63.  A  brick  is  8  in.  long,  4  in.  wide,  and  2  in.  thick.  How 
many  will  be  required  to  lay  ^  mile  of  8-foot  sidewalk,  the 
bricks  being  laid  flat  ? 

64.  How  many  will  be  needed  if  the  bricks  are  laid  with 
their  sides  uppermost  ? 

65.  How  much  larger  is  a  square  20  rods  in  diameter  than 
a  circle  of  the  same  diameter  ? 

66.  How  many  tons  of  ice  can  be  cut  from  ^  of  an  acre  of  a 
pond,  the  ice  being  12  inches  thick,  and  a  cubic  foot  weigh- 
ing 930  ounces  ? 

67.  How  many  square  yards  in  a  piece  of  calico  30  inches 
wide  and  45  yards  long  ? 

68.  I  gave  $  100  for  a  hektoliter  of  wine,  and  sold  it  for  $  5 
a  gallon.     Did  I  gain  or  lose,  and  how  much  ? 

69.  I  have  a  piece  of  land  150  feet  deep  and  fronting  200 
feet  on  the  street.  I  cut  it  into  50-foot  lots,  150  feet  deep, 
and  fence  each  lot.  What  does  my  fencing  cost  at  $  O.lGJy 
per  rod  ? 

70.  How  many  times  will  the  5-foot  driving-wheel  of  a  loco- 
motive revolve  in  going  from  Boston  to  Lawrence,  26  miles  ? 

"]  71.  How  many  cords  of  wood  can  be  piled  in  a  shed  40  ft. 
long,  12  ft.  wide,  and  8}  ft.  high  ? 

72.  How  many  square  feet  in  a  lot  of  land  1  rd.  8  ft.  long 
and  J  rd.  wide  ? 


REVIEW.  165 

73.  How  much  oil-cloth  8  feet  wide  is  needed  for  a  hall 
55  ft.  long  and  32  ft.  8  in.  wide  ? 

74.  How  many  acres  in  a  street  60  feet  wide  and  |  of  a  mile 
in  length  ? 

75.  At  $0.16f  per  square  yard,  what  will  it  cost  to  plaster 
a  room  20  ft.  square  and  12  ft.  high,  allowing  ^  for  openings  ? 

76.  What  will  the  flooring  of  the  above  room  cost  at  $  35  a 
thousand  feet,  j\  of  the  quantity  bought  being  wasted  ? 

77.  (17.28  -  0.083i)  X  3^|o  =  ?        ,  S'/  I        ^ 

78.  What  decimal  of  a  day  has  gone  at  3  p.  M.  ?        {•:>  2^^ 

79.  A  room  8  feet  high  is  16  feet  long  and  14  feet  wide. 
How  many  yards  of  paper  2  feet  wide  will  cover  the  walls  ? 

80.  A  steamer  reaches  Boston  at  3.15  p.  m.,  Aug.  3,  1881, 
by  the  steamer's  time,  after  a  voyage  of  7  d.  15  h.  30  min. 
Find  the  exact  time  of  her  sailing. 

81.  How  many  square  yards  of  silk  in  300  feet  of  3-inch 
ribbon  ? 

82.  What  cost  18  gal.  3  qt.  1  pt.  at  %  0.45  per  gallon  ? 
'        83.   Find  the  difference  between  0.7|  and  0.7  +  |. 

84.    From  an  acre  of  land  I  sold  8  square  rods,  and  also  a 
piece  8  rods  square.     How  much  have  I  left  ? 
"7^  85.   A  pile  of  wood  containing  4 J  cords  is  5  ft.  6  in.  high 
and  4 J  feet  wide.    How  long  is  it  ? 

86.  A  room  30  feet  long  requires  80  yards  of  carpeting  J  yd. 
wide.     How  wide  is  the  room  ? 

87.  A  field  containing  5  acres  is  600  feet  long.  Required 
the  distance  around  it. 

88.  The  ^^  desks  in  a  scbool-room  are  2  feet  by  18  inches. 
How  many  yards  of  40-ini'h  cloth  will  cover  them? 

89.  How  many  2-inch  cubes  will  a  cubic  yard  make  ? 

90.  How  many  feet  of  inch  boards  can  be  sawed  from  a 
stick  of  mahogany  15  feet  long  and  20  inches  square,  0.05  be- 
ing wasted  in  sawing  ? 


166  REVIEW. 

91.  From  a  lot  90  rods  square  I  sold  90  square  rods.  What 
is  the  value  of  the  remainder  at  $  240  per  acre  ? 

92.  From  a  piece  of  cloth  containing  9|  yards,  3f  yards 
were  cut.     What  part  of  the  whole  piece  remained  ? 

93.  A  garden  is  12  rods  long  and  10  rods  wide.  At  75 
cents  a  square  yard,  what  will  a  concrete  walk  6  feet  wide,  and 
Bxtending  around  the  garden  inside  the  fence,  cost  ? 

94.  A  garden  240  feet  long  and  160  feet  wide  is  enclosed 
by  a  tight  board  fence  6  ft.  high.  What  will  it  cost  to  paint 
both  sides  of  the  fence  at  5  cents  per  square  yard  ? 

95.  How  many  days,  hours,  and  minutes  from  10.30  p.  m., 
Feb.  4,  to  3.40  a.  m..  May  11  ? 

96.  A  regiment  of  900  soldiers  is  to  be  clothed ;  each  suit 
requires  9  yards  of  cloth  1^  yd.  wide.     How  many  yards  of  ^ 
flannel  |  yd.  wide  will  be  required  to  line  the  suits  ? 

97.  I  have  a  room  to  carpet  that  is  30  feet  long  and  27  feet 
wide.  Which  is  cheaper,  to  buy  yard-wide  carpeting  at  $  1.25, 
or  carpeting  f  yd.  wide  at  $  1,  and  how  much  ? 

98.  If  the  weight  of  a  hektoliter  of  wheat  is  75  kilos,  what 
weight  of  wheat  in  metric  tons  will  fill  a  bin  2  meters  long, 
1.4  meters  wide,  and  1  meter  deep  ? 

99.  What  will  it  cost  to  shingle  a  house  112  ft.  long,  each 
side  of  the  pitch-roof  being  25  ft.  wide,  10  shingles  covering  a 
square  foot,  the  shingles  costing  $  6.50  per  thousand. 

100.  Bought  7|J  dozen  at  18J  cents  a  dozen.  Required 
the  cost  of  12  dozen. 

101.  Sold  145  tons  of  coal  at  $  5f  per  ton,  and  12 J  tons  at 
$6J  per  ton,  and  charged  25  cents  a  ton  for  housing  the  coal. 
Find  the  amount  of  my  bill. 

102.  From  the  product  of  3J  and  4J  take  half  their  dif- 
ference. 

103.  If  17^  gallons  cost  $  19g,  what  will  |  of  a  gallon  cost  ? 

104.  From  a  piece  containing  J  of  a  yard  I  sold  J  of  a  yard 
at  $  I  a  yard,  and  gained  $  i.     What  did  the  piece  cost  me  ? 


H- 


PERCENTAGE.  167 


PEBCENTAGE. 

232.  1.  What  is  tW  of  100?    ^g^  ?    ji^?    ^^^?    ^^\? 

2.  What  is  T-3_o  of  500  ?     jU  ?     AV  ?     1^0%  ?     AV  ? 

3.  A  farmer  lost  15  sheep  from  a  flock  of  100.  How  many 
hundredths  of  the  flock  did  he  lose  ? 

4.  How  many  hundredths  of  $  1  are  17  cents  ?  39  cents  ? 
50  cents  ?     ^  of  a  cent  ? 

5.  How  many  hundredths  of  anything  is  I  of  it  ?     J  of  it  ? 

6.  In  a  catch  of  100  fish  f  were  perch.  How  many  hun- 
dredths  of  the  whole  were  perch  ? 

233.  Per  cent  means  hy  the  hundred.     Thus, 

3  per  cent  means  3  of  every  100,  or  3  hundredths. 

234.  The  Sign  %  is  used  for  the  words  per  cent.    Thus, 
3  %  means  3  per  cent,  and  4J  %  means  4|-  per  cent. 

235.  Percentage  treats  of  computing  in  hundredths. 

236.  The  Rate  per  cent  is  the  number  of  hundredths. 

237.  The  Base  is  the  number  of  which  the  hundredths 
are  taken. 

238.  The  Percentage  of  a  number  is  the  part  of  it 
denoted  by  the  rate  per  cent. 

239.  The  Rate  per  cent,  being  a  number  of  hundredths, 
is  a  fraction,  and  may  be  expressed  in  the  form  of  a  deci- 
mal or  of  a  common  fraction.     Thus, 


5%  =0.05    =rh  =  2\ 
12|  %=  0.121  =  f  21=  ^ 

25  /.  =  0.25    =^io-i 


621%  =  0.621-=  in  =  I 
100  %  =  1.00    =  -i-ff  =  1 

1371/.  =  1.371  =  -\U^  =  ¥ 
225  %  =  2.25    =  If-I  =  I 


168  PERCENTAGE. 

EXERCISES. 

Express  decimally : 

7.  5  %  ;  6  /. ;  7  %.  10.   ^^  %  ;  f  %  ;  IJ  %  ;  .07  %. 

a    J  %  5  2i  %  ;  8J  %.  11.   50  %  ;  i  %  ;  5  %  ;  .05  %. 

9.    12  %  5  125  %  ;  200  %.       12.    |  %  ;  60  %  ;  6  %  ;  600  %. 

Express  as  common  fractions  in  smallest  terms : 

la    10  %  ;  12J  %  ;  25  %.       16.   m^  %  ;  75  %  ;  80  %. 

14.  33  J  %  5  37|  %  ;  50  %.     17.   125  %  ;  150  %  ;  200  %. 

15.  16|  %  ;  621  %  .  831  %.  is.   40  %  ;  60  %  ;  14f  %. 

Change  the  following  fractions  to  hundredths : 

19-  i;  i;  i;  i;  *;  I;  i;  i- 

22.  f ;  f ;  f;  y^oJ  A;  #tj;  ?%;  ^(7- 

23.  f;  2%;  t;  |;  f;  i;  |;  |;  H- 
24-    if?  J  iftr  5   A"  j  "/r  5  T4  ?  "sV- 

The  Base  and  the  Rate  given,  to  find  the  Percentage. 
ORAL   EXERCISES. 

25.  What  is  5  %  of  $  4  ? 

Solution,  —  As  5  %  is  -^^j  5  %  of  $  4  is  yIij-j  ^^  ^o>  ^^  ^  "^j  which 
is  $0.20. 

26.  What  is  4  %  of  $40?     6  %  of  |44  ?     3  %  of  $400? 

27.  What  is  8  %  of  200  ?     Of  500  ?     Of  700  ?     Of  600  ? 

28.  What  is  10  %  of  90  yards  ?     12^  %  of  72  miles  ? 

29.  What  is  20  %  of  a  cubic  yard  ?     5  %  of  an  acre  ? 

30.  What  is  16§  %  of  a  day  ?     Of  a  foot  ?     Of  a  gross  ? 

31.  What  is  I  %  of  a  ton  ?     75  %  of  it  ?     |  of  it  ? 

32.  How  many  square  inches  in  66^  fo  oi  sl  square  foot  ? 

33.  What  is  the  difference  between  ^  of  a  mile  and  i  %  of 
a  mile  ? 


PEKCENTAGE.  169 

240.  The  Amount  is  the  base  plus  the  percentage. 

241.  The  Difference  is  the  base  less  the  percentage. 

WRITTEN    EXERCISES. 

34.  What  is  37^  ^  of  $  8.24  ? 

$  8.24  X  0.37i  =  $  3.09  Solution,  —  As  37^  %  is  0.37J, 

)j,  or  I,  37i%  of  $  8.24  is  0.37^  times 

1.0  3  $8.24,  or  I  of   $8.24,   which    is 

I^.^^X  1  =  13.09  $3.09. 

35.  A's  money  is  $  2575.  If  by  trading  he  should  increase 
it  by  40  %,  how  much  would  he  then  have? 

36.  A's  income  is  11890.  If  he  should  spend  83  J  %  of  it, 
how  much  would  he  have  left  ? 

242.     Rule  for  finding  the  Percentage. 

Multiply  the  base  by  the  rate  per  cent. 

Let  h  represent  the  base,  r  the  rate  per  cent,  'p  the  percentage,  a  the 
amount,  and  d  the  difference,  we  have  the 

Formulas  :  p  =  b  X  r,     a  =  b  -\-  p,     d  —  b  —  p, 

37.  Eind  2^  %  of  8500  tons.  41.  Find  9  J  %  of  $  5000. 

38.  Find  3  %  of  6840  gal.  42.  Find  8  %  of  $  645.50. 

39.  Find  f  %  of  2584  miles.  43.  Find  6  %  of  $  13.56. 

40.  Find  35  %  of  3460  men.  44.  Find  ^2\  %  of  $817.68. 

45.  Johnson  bought  a  house  for  $4850,  and  paid  down 
15  % .     How  much  did  he  then  owe  for  it  ? 

46.  A  merchant  sold  goods  which  cost  him  $  9675.75,  at  a 
profit  of  16  %.     How  much  did  he  get  for  the  goods  ? 

47.  Bought  a  bill  of  goods  amounting  to  $  186.80,  and  for 
cash  payment  obtained  a  deduction  from  it  of  5%.  How 
much  was  the  deduction  ? 

48.  How  many  persons  are  engaged  in  agriculture,  when 
they  constitute  24  %  of  a  population  of  61450  ? 


170  PERCENTAGE. 

49.  How  much  bank  currency  could  be  bought  for  $  4500, 
in  coin,  in  1864,  when  gold  was  at  285  %  ? 

50.  A  man  having  $  8550  bequeathed  33 J  %  to  his  wife  and 
the  remainder  to  his  children.  How  much  did  he  give  his 
children  ? 

51.  A  bought  goods  to  the  value  of  $  345.75,  and  sold  them 
to  B  at  15  %  advance  on  his  outlay,  and  B  sold  them  to  C  at 
15  %  less  than  his  outlay.     How  much  did  C  give  for  them  ? 

Base  and  Percentage  given,  to  find  the  Rate  per  cent 
ORAL   EXERCISES. 

52.  What  per  cent  of  12  is  3  ? 

Solution.  —  3  is  y%  of  12  ;  and  ^  =  ^,  or  0.25,  or  25  %. 

53.  What  per  cent  of  $  35  is  $  7  ?     Is  $  14  ?     Is  $  28  ? 

54.  What  per  cent  of  24  is  6  ?     Is  12  ?     Is  18  ? 
What  per  cent 

55.  Of  42  miles  is  21  miles  ?  58.   Of  $  6  is  $  4.50  ? 

56.  Of  75  tons  is  45  tons  ?  59.   Of  $  64  is  $  40  ? 

57.  Of  48  pounds  is  30  pounds  ?     60.   Of  $  96  is  $  36  ? 

61.  If  4  quarts  of  grain  are  given  for  grinding  a  bushel, 
what  per  cent  is  the  cost  of  grinding  ? 

62.  John  earns  each  week  $7.50.  He  spends  for  board 
1 2.50,  and  as  much  more  for  other  things.  What  per  cent 
of  his  earnings  are  his  spendings  ? 

WRITTEN    EXERCISES. 

63.  What  per  cent  of  128  is  16  ? 

16  -i-  128  =  0.12^,  or  12^ %  Solution,—  16  is  ^  of  128. 

Or,  As  i^i=  .12^,  or  12^%,  16  is 

^  =  i  =  .12i,  or  12i  %  12i  %  of  128. 

64.  What  per  cent  of  600  bushels  is  57  bushels  ? 


PERCENTAGE.  171 


243.     Rule  for  finding  the  Rate  per  cent. 

mtage  by  the  base,  extern 

Formula,     t  —^  ^h. 


Divide  the  percentage  by  the  base,  extending  the  division  t:* 
hundredths. 


What  per  cent  is 

65.  182.75  of  2150  ?  69.   600  of  720  ? 

66.  $  8  of  $  130  ?  70.   70  tons  of  500  tons  ? 

67.  16  bu.  of  62  J  bu.  ?  71.   $  85  of  $  1700  ? 

68.  $490  of  $5000?  72.   $  57.375  of  $  765  ? 

73.  Of  a  farm  containing  1640  acres  there  are  246  acres  in 
meadow.     What  per  cent  is  in  meadow  ? 

74.  If  the  income  on  $  1346  is  %  168.25,  what  is  the  rate  ? 

Hate  and  Percentage  given,  to  find  the  Base. 
ORAL    EXERCISES. 

75.  $  25  is  5  %  of  my  money.     How  much  have  I  ? 

Solution.  —  As  $  25  is  5  %  of  my  money,  1  %  of  my  money  is  \  of 
$  25,  or  ^  5 ;  100  %  of  my  money  must  be  100  times  $  5,  or  $  500.    Or, 

As  $25  is  5  %,  or  -^^  of  my  money,  |^  of  my  money  must  be  20 
times  $  25,  or  $  500. 

76.  $  6  is  20  %  of  what  number  ?  40  rods  is  12  J  %  of  what 
number  ? 

77.  1  pound  4  ounces  is  25  %  of  how  many  pounds  ? 

78.  A  number  increased  by  25  %  of  itself  is  60.  What  is 
the  number  ? 

Solution.  — As  a  number  increased  by  25  %  of  itself  must  be  125  %, 
or  I  of  itself,  60  must  be  |  of  the  number,  and  f ,  or  the  number, 
must  be  48. 

79.  Sold  a  watch  for  $  55,  which  was  10  %  above  its  cost. 
What  was  its  cost  ? 

80.  42  is  33 J  %  less  than  what  number?  33J%  more  than 
what  number  ? 


172  PERCENTAGE. 

WRITTEN    EXERCISES. 

81.   $  3550  is  25  %  of  what  number  ? 

^4  ^  ^  .^^^  Solution.  —  As    $  3550    is 

$  ^11^  X  ;pp  =  114200  25  f„    of  an  unknown  num- 

ber,    1  %     of    the    unknown 


Or,  $3550 

4 


number  is  -^-^  of  $  3550,   oi 

$  a|M. ;  100  %  of  the  unknown 

$14200  number   must   be    100   times 

$  ^f  ^  or  $  14200. 

Or,  as  $  3550  is  25  %,  or  J  of  the  number,  the  number  must  be  four 

times  $  3550,  or  $  14200. 

82.  $  16.22  is  5  %  of  what  number  ? 

83.  Having  sold  40  %  of  my  sheep,  I  have  177  left.  What 
number  had  I  at  first  ? 

244.     Rule  for  finding  the  Base. 

Divide  the  percentage  by  the  rate  per  cent, 

Formulas,     h  =p  -^  r.    &  =  a-f-(l4-r).    b  =  d^  {1  —  r), 

Find  the  number  of  which 

84.  $  125  is  8  %.  87.  7.80  yd.  is  |  %. 

85.  0.108  of  a  ton  is  f  %.  88.  21.6  rd.  is  f  %. 

86.  16  bu.  is  24  %.  89.  $  14  is  ^  %. 

90.  I  burned  1750  lb.  of  coal,  or  25  %  of  my  supply,  in  a 
single  week.     How  much  had  I  ? 

91.  The  expenses  of  a  charity  concert  were  40  %  of  the 
receipts.    The  poor  received  $  250.    What  were  the  expenses  ? 

92.  An  agent's  salary  having  been  decreased  33 J  %  is  now 
$  1600.     What  was  it  at  first  ? 

93.  A  man  owes  $  6750,  which  is  75  %  as  much  as  he  is 
worth.     How  much  is  he  worth  ? 

94.  The  voters  of  a  certain  city  number  16386,  wliich  is 
20  %  more  than  the  number  3  years  ago.  What  was  the 
number  then  ? 


PERCENTAGE.  173 

PROFIT   AND    LOSS. 

/ 

95.  At  a  gain  of  20  %  of  the  cost  of  an  article,  what  part  of 
the  cost  equals  the  gain  ? 

96.  How  much  is  gained  by  selling  goods  at  10  fo  profit 
when  the  cost  is  $  25  ? 

97.  How  much  is  lost  on  goods  which  cost  %  40  by  selling 
them  at  a  loss  of  V2i\  %  ? 

98.  A  grocer  buys  tea  at  45  cents  a  pound,  and  sells  it  at 
60  cents  a  pound.     What  is  the  gain  per  cent  ? 

99.  Sold  a  cow  which  cost  me  %  40  for  %  45.  What  was 
the  gain  per  cent  ? 

100.  Sold  a  horse  for  %  120,  and  gained  16§  %.  What  did 
it  cost  ? 

101.  Sold  goods  for  %  108,  and  lost  10  %,     What  was  their 

cost? 

245.  Profit  and  Loss,  as  commercial  terms,  express  the 
gain  or  loss  in  business  transactions. 

246.  The  hase  of  computation  is  the  cost,  the  gain  or  loss 
is  the  "percentage, ;  and  the  cases  which  occur  correspond 
to,  and  are  solved  like  the  preceding  cases  of  percentage. 

WRITTEN    EXERCISES. 

102.  How  much  is  made  by  selling  flour  at  20  %  profit 
which  cost  %  7.75  a  barrel  ? 

103.  A  dealer  sold  a  stock  of  goods,  damaged  by  fire,  at  an 
average  of  66f  %  less  than  cost.  The  cost  being  %  18240,  what 
was  the  loss  ? 

104.  Of  a  carload  of  fruit  25%  is  spoiled.  I  sell  the  re- 
mainder for  16f  %  advance  on  its  cost.  What  is  my  profit  or 
loss  on  the  load  if  it  cost  %  400  ? 


174  PERCENTAGE. 

105.  A  merchant  sold  a  cask  of  molasses  whicli  cost  him 
$  69.60  at  a  profit  of  15  %.     What  did  he  gain  ? 

106.  If  I  pay  for  goods  $  350.50,  and  sell  them  at  6  J  %  loss, 
how  much  shall  I  lose  ? 

107.  Sold  a  house  which  cost  me  $  3500  at  a  gain  of  8  %. 
What  did  I  receive  for  it  ? 

108.  Sold  a  carriage  for  $  210  at  a  loss  of  16  %.  What  was 
the  cost  ? 

109.  What  per  cent  was  gained  hy  selling  property  that 
cost  $2400  for  $2550? 

110.  Bought  hams  for  8^  cents  a  pound.  What  per  cent 
will  be  gained  by  selling  them  at  12  cents  a  pound  ? 

111.  Sold  goods  for  $  3312.70,  and  made  5 J  %.  What  was 
the  cost  ? 

112.  If  I  am  compelled  to  lose  12j^%  on  damaged  goods,  how 
should  I  sell  those  that  cost  me  $  560  ? 

113.  When  the  cost  of  flour  is  $  7.50  a  barrel,  and  the  ex- 
pense of  selling  6  %,  at  what  price  must  it  be  sold  to  gain  5  %  ? 

\^  114.  Of  goods  worth  $  1600,  one  fourth  is  sold  at  a  profit  of 
15  %.  For  how  much  must  the  remainder  be  sold  to  gain  20  % 
on  the  whole  ? 

115.  Sold  a  watch  for  $28,  and  gained  12%.  What  per 
cent  would  I  have  gained  or  lost  if  I  had  sold  it  at  $  24  ? 

116.  Bought  a  cask  of  molasses  containing  120  gallons  for 
$  50.  But,  a  fifth  of  the  molasses  having  leaked  out,  for  what 
must  the  remainder  be  sold  a  gallon  to  gain  10  %  on  the  pur- 
chase ? 

117.  By  selling  hay  at  $15  a  ton  I  lose  10%.  At  what 
price  must  I  sell  it  to  gain  15  %  ? 

118.  A  merchant  sold  goods  for  $  150  and  lost  10  %,  whereas 
he  should  have  gained  30  %  per  cent.  How  much  were  they 
sold  under  their  proper  value  ? 

119.  Bought  goods  for  $  14500.  Half  of  them  I  am  obliged 
to  sell  at  a  loss  of  20%.  If  I  sell  the  other  half  at  a  gain  of 
20  %,  what  shall  I  gain  or  lose  on  the  whole  ? 


PEKCENTAGE.  175 


COMMISSION. 


120.  How  much  should  be  received  for  selling  $  500  worth 
of  goods  if  3  %  is  allowed  ? 

121.  How  much  must  be  paid  for  selling  $  800  worth  of 
goods  at  5  %  ? 

122.  A  collector  of  $  700  was  paid  2 J  %  for  his  services. 
How  much  did  he  receive  ? 

247.  Commission,  or  Brokerage,  is  the  compensation  or 
percentage  allowed  an  agent  for  transacting  business. 

The  agent  may  be  known  as  a  factor,  broker,  or  commis- 
sion merchant. 

248.  The  commission  is  usually  a  certain  per  cent  of 
the  sum  invested  or  collected  by  the  agent,  the  investment 
or  the  collection  being  the  hase,  and  the  commission  the 
percentage. 

WRITTEN    EXERCISES. 

123.  What  is  the  commission  on  the  sale  of  $  5678  worth  of 
cotton  cloth  at2i%? 

124.  A  broker  negotiated  the  sale  of  $  3500  United  States 
securities  at  a  brokerage  of  \  %.     What  was  the  brokerage  ? 

125.  A  commission  merchant  sold  goods  to  the  amount  of 
$  7896.50.     What  was  his  commission  at  2  %  ? 

126.  Pind  the  commission  on  the  sale  of  368  barrels  of  flour 
for  $  6.50  a  barrel,  the  rate  of  commission  being  2J%. 

127.  An  agent  invests  $  5000  for  me  in  the  purchase  of 
land.  His  commission  being  \  %,  how  much  shall  I  send  him 
to  cover  all  the  cost  ? 

128.  My  agent  has  sold  goods  for  me  amounting  to  $  13500. 
His  charges  are  :  commission,  2^  %  f  guarantee,  2  %  ;  cartage 
and  storage  $  16.50.     How  much  is  due  me  ? 


1 76  PERCENTAGE. 

129.  I  sent  my  agent  $  1500  to  invest  after  deducting  his 
commission  of  2J  %.  Wliat  sum  can  he  invest,  and  what  will 
be  his  commission  ? 

Solution.  —  The  remittance  includes  the  investment  and  the  com- 
mission. The  investment  is  100  %  of  the  investment,  and  the  com- 
mission is  2-|  %  of  the  investment.  The  remittance,  therefore,  or 
$1500,  must  he  102^%  of  the  investment.  $1500  is  102^%  of 
I J  ^^^-  X  1  0  0 ,  or  $  1463.41 +.  This,  subtracted  from  the  remittance, 
$  1500,  gives  as  the  commission  $  36.59. 

130.  A  commission  merchant  receives  $  650  to  invest  in 
goods  after  deducting  his  commission  of  3  %.  What  will  be 
his  commission  ? 

131.  I  have  remitted  to  an  agent  $1426.80,  with  instruc- 
tions to  lay  it  out  for  me  in  flour  at  $  6.50  a  barrel,  after  de- 
ducting his  commission  of  2^  %.  How  many  barrels  can  he 
buy? 

132.  An  agent  received  a  sum  of  money  to  lay  out  after  de- 
ducting his  commission  of  2J  %.  He  laid  out  $  1392.  What 
was  the  sum  he  received  ? 

133.  A  commission  merchant  sold  on  commission  goods  for 
i  8134.75,  and  received  $334.75,  which  included  a  charge  for 
cartage,  freight,  and  storage  of  $  22.75.  What  was  the  rate 
of  commission  ? 

134.  A  lawyer  collected  65%  of  a  note  of  $  950,  and  charged 
6J  %  commission.     Find  his  commission. 

135.  A  factor  received  5  %  for  buying  wool.  His  commis- 
sion amounted  to  $208.50.  How  much  did  he  pay  for  the 
wool  ? 

136.  Sent  my  agent  $4100  for  the  purchase  of  iron  after 
taking  out  his  commission  of  2 J  %.  After  he  had  bought  the 
iron,  I  changed  my  business  and  telegraphed  him  to  sell 
the  iron  at  cost.  He  did  so,  taking  a  commission  of  2^  %  on 
the  sale,  and  sent  me  the  balance.  How  much  did  I  lose  by 
the  transa(;tion  ? 


■^ 


PERCENTAGE.  177 

INSURANCE. 

137.  What  is  the  cost  of  securing  the  payment  to  me  of 
$  1000  in  case  my  house  is  destroyed  by  fire,  when  2  %  is  paid 
to  those  taking  the  risk  ? 

138.  What  will  be  the  annual  charge  at  1^  %  for  taking  the 
risk  of  $  3000  against  loss  or  damage  on  merchandise  ? 

249.  Insurance  is  security  against  loss. 

250.  The  Premium  is  the  sum  paid  for  insurance. 

251.  The  Policy  is  the  writing  containing  the  contract 
of  the  insurer  with  the  insured. 

252.  The  sum  insured  is  the  hase,  and  the  premium  is 
the  percentage. 

WRITTEN    EXERCISES. 

139.  What  is  the  cost  of  insuring  $  3600  on  a  house  for  5 
years  at  2  %,  and  $  1  for  the  policy  ? 

140.  What  is  the  premium  for  insuring  $  5545  on  merchan- 
dise for  one  year  at  2 J  %  ? 

141.  What  is  the  amount  paid  for  insurance  on  |  of  a  ship 
valued  at  $  68000  at  3  %,  and  $  1  for  the  policy  ? 

142.  A  factory  is  insured  for  $  55000  at  2^  %.  If  the  prop- 
erty should  be  burned,  what  loss  would  the  insurer  actually 
sustain  ? 

143.  A  house  was  insured  for  |  of  its  value  at  1^  %,  and 
the  premium  was  $  27.    What  was  the  value  of  the  house  ? 

144.  Paid  $  73,  including  $  1  for  policy,  for  the  insurance 
of  $  3600  in  a  house.     What  was  the  rate  of  insurance  ? 

145.  A  man  44  years  of  age  takes  out  a  life-policy  for 
$  15000  for  the  benefit  of  his  wife,  at  the  yearly  rate  of  $  26.50 
per  $1000.  Should  his  death  occur  at  the  age  of  74,  how 
much  more  would  his  widow  receive  than  had  been  paid  in 
yearly  premiums  ? 


178  PERCENTAGE. 

MISCELLANEOUS    EXERCISES. 

146.  The  sum  of  |  %,  5  %,  24  %  and  55  %  of  a  number  is 
60.25.     What  is  the  number  ? 

147.  An  article  is  composed  of  37  parts  of  pure  silver  and 
3  parts  of  copper.  What  per  cent  of  the  whole  is  each  of  the 
components  ? 

148.  A  clerk  receiving  a  salary  of  $  950  pays  $  275  a  year 
for  board,  $  180  for  clothing,  and  $  150  for  other  expenses. 
What  per  cent  of  his  salary  is  left  ? 

149.  I  bought  150  apples  at  2  for  a  cent,  and  150  at  3  for  a 
cent.  I  sold  them  all  at  5  for  2  cents.  How  much  did  I  gain 
or  lose  ? 

150.  A  man  bought  8  books  at  the  rate  of  $  10  a  dozen,  and 
sold  them  for  $  1.75  each.     What  per  cent  was  gained  ? 

151.  Bought  a  bill  of  goods  amounting  to  $  1540,  but  a 
discount  of  25  %  was  allowed  with  5  %  off  for  cash.  How 
much  did  I  pay  ? 

152.  If  I  should  forward  %  603.75  to  a  broker  in  St.  Louis 
for  the  purchase  of  flour,  his  brokerage  being  5%,  how  many 
barrels  of  flour  should  he  return  at  $  5  a  barrel  ? 

153.  Lost  $  17.25  by  selling  a  watch  15  %  below  cost.  What 
was  the  cost  ? 

154.  For  what  sum  must  a  store  and  its  goods,  valued  at 
$  25640,  be  insured  so  as,  in  case  of  its  destruction,  to  recover 
the  entire  value  of  the  building  and  goods  and  the  premium 
of  2  %  ? 

155.  If  I  purchase  15  pieces  of  cloth,  of  35  yards  each,  at 
%  4.25  a  yard,  for  how  much  a  yard  must  I  sell  the  whole  so 
as  to  gain  25  %  ? 

156.  A  man  bought  80  tons  of  coal  at  $  5  a  ton,  10  %  of 
which  went  down  in  a  boat  and  was  lost.  For  how  much  a 
ton  must  the  remainder  be  sold  that  he  may  lose  nothing  ? 

157.  A  man  bought  4500  bushels  of  wheat  at  %  1.20  a  bushel. 
He  sold  10%  of  it  at  3  %  loss,  50%  of  it  at  10  %  gain,  and  the 
remainder  at  5%  gain.  What  was  gained  by  the  entire  trans* 
action  ? 


+ 


PERCEITTAGE.  179 

158.  If  you  should  sell  a  house  for  $  6000,  and  lose  33 J  %, 
for  what  should  you  sell  another  at  the  advance  of  32  %  for  just 
enough  to  cover  the  loss  upon  the  first  house  ? 

159.  In  1875  there  were  104513  illiterate  persons  in  Massa- 
chusetts out  of  a  population  of  1652000.  What  was  the  rate 
per  cent  of  illiteracy  ? 

160.  I  receive  a  remittance  of  $  13195  to  be  spent,  after 
paying  commission  of  1^  %,  in  the  purchase  of  coal.  Eequired 
my  commission. 

161.  My  house  cost  me  $  12000.  It  is  insured  for  |  of  its 
value  at  |%  premium.  What  is  my  actual  loss  in  case  it 
burns  ? 

162.  The  premium  on  an  insurance  of  I  930  was  $  23.25. 
What  was  the  rate  ? 

163.  An  agent  collects  money  at  2^%,  and  pays  his  prin- 
cipal $  4387.50.     What  was  the  amount  of  the  collection  ? 

164.  Leonard  &  Co.  sell  a  lot  of  goods  for  me  at  auction  to 
the  amount  of  $  11500.  Their  charges  are  as  follows :  com- 
mission, 2J%;  guarantee,  2^%  ;  advertising,  $35;  labor  and 
storage,  $  17.25.     How  much  is  due  me  ? 

165.  Sold  property  for  $1400,  25%  of  which  is  gain.  I 
found  myself,  however,  able  to  collect  only  90  %  of  the  pro- 
ceeds of  the  sale.     What  was  my  actual  gain  %  ? 

166.  Giles  lost  8|  %  of  his  money  in  speculation,  but  had 
$  920  left.     How  much  had  he  at  first  ? 

167.  My  room  is  24  feet  long;  its  width  is  50%  of  its 
length ;  how  many  yards  of  carpeting,  J  yd.  wide,  will  it  re- 
quire ? 

168.  A  grocer  after  losing  11  %  of  his  apples  has  133.5  bar- 
rels left ;  if  they  cost  him  $  2.50  per  barrel,  for  what  must 
they  be  sold  that  he  may  lose  nothing  on  his  purchase  ? 

169.  My  agent  sells  500  barrels  of  flour  at  $  10  per  barrel 
and  remits  me  $4750.  What  rate  of  commission  did  he 
charge  for  selling  ? 


180  PERCENTAGE. 

170.  My  store  is  insured  for  $  8000  at  1 J  %  premium,  and 
mj  stock  for  $  15000  at  f  %.  If  both  are  entirely  consumed, 
what  is  the  underwriters'  loss  ? 

171.  The  difference  between  24  %  and  55  %  of  a  number  is 
60.45.     "What  is  the  number  ? 

172.  Bought  a  range  of  wood  20  ft.  long,  12  ft.  high,  and 
4  ft.  wide,  at  $  5  per  cord,  and  sold  the  whole  for  $  50.  Be- 
quired  the  per  cent  of  gain. 

173.  Paid  $30  for  my  winter's  wood,  which  was  to  have 
been  4  feet  in  length.  It  averaged,  however,  but  44  inches. 
Out  of  how  much  money  was  I  cheated  ? 

174.  In  a  school  of  400  scholars  there  were  120  absences 
in  4  weeks ;  the  school  has  2  sessions  5  days  in  a  week. 
What  was  the  per  cent  of  attendance  ? 

175.  How  must  I  mark  cloth  which  cost  $2.50  so  as  to 
gain  20  %  and  still  fall  25  %  from  my  marked  price  ? 

176.  A  cubic  foot  of  water  weighs  62f  pounds  and  a  cubic 
foot  of  ice  57 1-  pounds.  Ice  is  what  per  cent  lighter  than  an 
equal  bulk  of  water  ? 

177.  The  population  of  Chicago  in  1880  was  503620,  an  in- 
crease of  69  %  in  ten  years ;  what  was  the  city's  population  in 
1870  ? 

QUESTIONS. 

233.  What  does  per  cent  mean  ?  235.  Of  what  does  percentage 
treat?  236.  What  is  the  rate  per  cent  ?  237.  The  base?  238.  The 
percentage  of  a  number  1 

239.  How  may  the  rate  per  cent  be  expressed  ?  240.  What  is  the 
amount  ?     241.  The  difference  ? 

242.  The  base  and  rate  being  given,  how  is  the  percentage  found  ? 

243.  The  base  and  percentage  being  given,  how  is  the  rate  found? 

244.  The  rate  and  percentage  being  given,  how  is  the  base  found  ? 
245.  Define  profit  and  loss.     246.  What  is  the  base  of  computa- 
tion ?     247.  What  is  commission,  or  brokerage?      248.  What  is  the 
base  ?     The  percentage  ? 

249.  What  is  insurance  ?    250.  The  premium  ?    251.  The  policy  1 


INTEKEST.  181 


INTEREST. 

253.  1.  When  money  is  loaned  for  a  year  at  7%,  what 
part  of  the  money  is  the  per  cent  ? 

2.  How  much  must  be  paid  for  the  use  of  $  15  for  1  year  at 
57c?    At6%? 

3.  How  much  must  be  paid  for  the  use  of  $  20  at  7  %  for 
1  year  ?     For  2  years  ? 

4.  When  $  200  is  borrowed  for  2  years  at  7  %  a  year,  what 
amount  should  the  borrower  pay  at  the  end  of  that  time  ? 

254.  Interest  is  the  money  paid  for  the  use  of  money. 

255.  The  Principal  is  the  money  for  whose  use  interest 
is  paid. 

256.  The  Amount  is  the  sum  of  the  principal  and  the 
interest. 

257.  The  Rate  of  interest  is  the  number  of  hundredths 
of  the  principal  taken  as  the  interest  for  one  year  or  other 
specified  time. 

Note  1.  — The  rate  for  one  year  and  at  6  %  is  to  be  understood  in  this  book 
when  no  other  time  or  rate  is  specified. 

Note  2.  —  The  rate  of  interest  is  regulated  by  law.  The  legal  rates  in  the 
diiferent  States  may  be  found  in  a  table  in  the  Appendix. 

SIMPLE    INTEREST. 

258.  Simple  Interest  is  interest  on  the  principal  alone. 
Interest  is  an  application  of  percentage,  the  principal 

being  the  base,  the  annual  rate  multiplied  by  the  time  in 
years  being  the  rate  per  cent,  and  the  interest  the  per- 
centage. 

259.  In  the  computation  of  interest  it  is  customary  to 
consider  a  year  as  consisting  of  12  months  of  30  days  each. 


182  INTEREST. 


General  Method. 
ORAL   EXERCISES. 

5.  What  is  tlie  interest  of  $  50  for  1  year  at  4  %  ? 

Solution. — At  4  %  1  year's  interest  is  .04  of  the  principal,  and.  04.  jt 

$50  is  $2. 

6.  What  is  the  interest  of  $  60  for  1  year  at  5  %  ? 

7.  What  is  the  interest  of  $  200  for  1  year  at  6  %  ?     For 
3  years  ?     For  5  years  ? 

8.  What  is  the  interest  of  $  200  for  2  years  6  months  at 

7%? 

Solution.  —  As  at  7  %  the  interest  of  $  200  for  1  year  is  $  14,  for 
2  years  6  months,  or  2J  years,  it  must  be  2^  times  $  14,  or  $35. 

9.  What  is  the  interest  of   $  100  for  2  years  3  months  at 
8  %  ?     For  3  years  1  month  ? 

10.  What  is  the  amount  of  $  400  for  2  years  9  months  at 
6  %  ?     For  3  years  4  months  ? 

WRITTEN    EXERCISES. 

11.  What    is    the    interest    and   what    is    the    amount   of 
$26.25  for  2  years  4  months  at  7%? 

$  26.25  =  Principal 

.07  =  Eate  Solution. —  One  yearns   in- 

1 1.8375  =  1  year's  interest  terest   is   .07   of  $26.25,  or 

21  $  1.8375  ;    2^  years'  interest 

gj25~  is  2 J  X  S  1.8375,  or,  to  the 

oQjf^Q  nearest  cent,  $4.29.    Adding 

x-.-oo^K       T  i.        i.  the  principal  to  the  interest, 
1 4.2875  =  Interest  ,        .!  ^  dt  on  ^a 

r.^  r»^  -r.  .     .     1  we  have  the  amount,  $  30.54. 

26.25      =  Principal 


$30.54      =  Amount 

12.   What  is  the  interest  and  what  is  the  amount  of  $  1728 
for  3  years  9  months  at  6  %  ? 


INTEREST.  183 

13.  What  is  the  interest  of  $  144  for  1  year  8  months  at 

5%? 

14.  What  is  the  interest  of  $556  for  3  years  5  months 
7  days  at  8  %  ? 

$556 

.08 


$  44.48  Solution,  —  One  year's  interest  is  .08 

41.24  ^^  $^56,  or  1 44.48  ;  the  interest  for  3 

TTooi  years  5  months  7  days,  or  41. 2 J  months, 

8896  or  ^i|i  years,  is  ^^  times  $  44.48, 


4448  . 
17792 
12)  $  1834.0581 


or,  to  the  nearest  cent,  $  152.84. 


$  152.838+ 

15.  What  is  the  interest  of  $  720  at  5  %  for  1  y.  7  mo.  18  d.  ? 

$  720  Solution.  —  One  year's   interest  is    .05    of 

.05  $  720,  or  $  36  ;  the  interest  for  1  y.  7  mo.  18  d., 

12)  $36  00  ^^  ^^'^  ^^''  °^  ^^^^  ^^^^^'  ^^  ^^^  ^  *^^'  ^^ 

• '- —  $58.80.     In  this   example,  1   year's  interest 

$  3.00  being  a  multiple  of  12,  we  divide  by  the  de- 

1^'^  nominator  of  the  multiplier  before  multiplying 

$  58.80  by  the  numerator. 

16.  Find  the  amount  of  $  1500  for  2  y.  6  mo.  15  d.  at  6  %. 

260.    General  Rule  for  Interest. 

Multiply  the  principal  by  the  rate,  and  this  product  by  the 
time  in  years. 

To  find  the  amount,  add  the  principal  and  the  interest. 

Let  jp  represent  the  principal,  r  the  rate  per  cent,  i  the  interest, 
t  the  time,  and  a  the  amount,  and  we  have  the 

Formulas,    i  =p  xr  x  t       a  —p  +  %, 


184  INTEREST. 

Note.  —  In  interest  partial  results  may  be  carried  to  four  places  of  deci- 
mals. The  answers,  in  business  transactions,  are  deemed  sufficiently  exact  if 
the  mills  are  omitted,  and  when  they  are  five  or  more,  the  cents  are  increased 
byl. 

17.  What  is  the  interest  of  $  2464  for  2  y.  9  mo.  15  d.  at 

18.  What  is  the  interest  of  $  2503.75  for  3  y.  10  mo.  21  do 
at6%? 

19.  What  is  the  interest  of  $  560.50  for  4  y.  10  d.  at  7  %  ? 

20.  What  is  the  interest  of  $  97.16  for  1  y.  5  mo.  at  6  %  ? 

21.  What  is  the  interest  of  $156.80  for  3y.  1  mo.  3d.  at 
4%? 

22.  What  is  the  interest  of  $865  for  1  y.  9  mo.  24  d.  at  8%? 

23.  What  is  the  interest  of  $  890  for  5  y.  7  mo.  8  d.  at  6  %  ? 

24.  What  is  the  amount  of  $  5000  for  3  y.  11  mo.  10  d.  at 
7%? 

Six  per  cent  Method. 

261.  The  Interest  of  any  sum,  at  6  per  cent  a  year. 

For  12  months,  or    1  year,  is  .06  of  the  principal. 
For    2  months,  or    ^  year,  is  .01  of  the  principal. 
For    1  month,   or  30  days,  is  .00|-  of  the  principal. 
For    \  month,   or    6  days,  is  .001  of  the  principal. 
For  -^  month,   or    1  day,    is  .000 1  of  the  principal. 

262.  Hence,  as  a  convenient  method  of  reckoning  in- 
terest at  6  per  cent. 

Multiply  J  of  .01  of  the  pi'incijpal  by  the  time  in  months. 
Or, 

Multiply  .001  of  the  principal  hy  \  of  the  time  in  days. 
Or, 

Of  the  principal  take  6  times  as  many  hundredths  as 
years,  ^  as  many  hundredths  as  months,  and  J  as  many 
thoiosandths  as  days. 


INTEREST. 


185 


/  25.    What  is  the  interest  of  $  926  for  3  years  11  months 

15  days  at  6  %  ? 


2)  $926.    =  Principal 

$  4.63    =  1  mo.'s  interest 
47|  =  Time  in  months 
2311 
3241 

1852 


$  219.921  =  Eequired  interest     «  219.92J. 


Solution.  —  Two  months' 
interest  is  .01  of  the  prin- 
cipal ;  1  month's  interest  is 
^  of  .01  of  the  principal, 
$926,  or  $4.63;  the  interest 
for  3y.  11  mo.  15  d.,  or  47^ 
mo.,   is    47i    X    $4.63,   or 


26.   What  is  the  interest  of  $  340  for  103  days  at  6  %  ? 

$340    =  Principal 
$  0.34    =z  Six  days'  interest 
17J  =  J-  time  in  days 

238 
34 
$  5.83|  =  K-equired  interest 


Solution.  —  Six  days'  inter- 
est is  .001  of  the  principal, 
or  $0.34  ;  one  day's  interest 
is  ^  of  $0.34,  and  103  days' 
interest  is  103  X  i  of  $0.34, 
or  J^  of  $0.34,  or  17^  X 
$0.34,  or  $5.84,  to  the  near- 
est cent. 


27.   What  is  the  interest  of  $  1650  for  2  y.  7  mo.  18  d.  at 

6%? 


Int.  for  2j.  =  .12  of  principal.     $  1650 


"    7  mo.  =  .035 

"    18  d.  =  .003 

.158 


.158 
13200 
8250 
1650 


Solution,  —  Taking 
.06  for  each  year's  in- 
terest, .00 J  for  each 
month's  interest,  and 
.000 J  for  each  day's 
interest,  we  find  that 
the    interest    for    the 


$260,700 
given  time  is  .158  of  the  principal,  or  .158  of  $  1650,  or  $260.70. 

What  is  the  interest  at  6  %  of 

28.  $  56.80  for  1  y.  8  mo.  17  d.  ? 

29.  $  6000  for  4  y.  2  mo.  ? 

30.  $  17.28  for  1  y.  11  mo.  3d.? 


-/- 


186  INTEREST. 

31.  $  1850.75  for  9  mo.  24  d.  ? 

32.  $253.50  for  2  y.  4mo.  7d.? 

33.  $  85.90  for  3  y.  6  mo.  27  d.  ? 

34.  $  1992.25  for  93  days  ? 

35.  $  15600  for  4  y.  7  mo.  19  d.  ? 

36.  What  is  the  amount  of  $  1400  for  2  years  6  months  ? 

37.  What  is  the  amount  of  $  7000  for  5  years  3  months  ? 


263.  For  any  other  rate  than  6  per  cent  we  may,  when 
more  convenient  than  to  apply  the  general  rule  (Art.  260), 

Find  the  interest  at  6  ^er  cent,  and  increase  or  diminish 
this  interest  hy  such  part  of  itself  as  will  give  the  interest  at 
the  required  rate. 

That  is,  to  find  4  %  interest  subtract  from  the  6  %  inter- 
est J  of  itself ;  4 J  %  interest,  subtract  from  the  6  %  interest 
\  of  itself  ;  5  %  interest,  subtract  from  the  6  %  interest  \  of 
itself ;  7  %  interest,  add  to  the  6  %  interest  \  of  itself ;  and 
so  on. 

38.  What  is  the  interest  of  $  545  for  8  mo.  24  d.  at  4%  ? 

39.  What  is  the  interest  of  $  78.50  for  123  days  at  5  %  ? 

40.  What  is  the  interest  of  $  64.70  for  2  y.  5  mo.  at  7  %  ? 

41.  What  is  the  interest  of  $  1440  for  11  mo.  23  d.  at  ^%? 

42.  What  is  the  interest  of  $  9500  for  3  y.  6  mo.  17  d.  af 
7%? 

43.  What  is  the  interest  of  $  600.80  for  2  y.  11  mo.  3  d 
at8%? 

44.  What  is  the  interest  of  $  20000  for  63  days  at  5  %  ? 

45.  What  is  the  interest  of  $340.90  for  4y.  7  mo.  lid. 
at7%? 

46.  What  is  the  interest  of  $  15420  for  9  mo.  24  d.  at  6i  %  ? 

47.  What  is  the  interest  of  $  374.75  for  3  y.  9  rao.  at  8  %  ? 


INTEREST.  187 

4a  Ernest  Williams  borrowed,  April  5,  1881,  $525  at  7% 
interest,  and  kept  it  until  May  16,  1882.  What  was  the 
amount  ? 

49.  A  note  for  $  450.60,  dated  March  5,  1880,  was  paid 
Dec.  31,  1881,  with  interest  at  7  %.     What  was  the  amount  ? 

Short  Method. 

264.  The  following  method  of  computing  6%  interest 
is  often  very  convenient,  especially  for  short  times. 

Find  60  days'  interest  hy  taking  .01  of  the  principal. 
Then  take  such  multiples  or  parts  of  this  interest  as  the  given 
time  may  require, 

50.  What  is  the  interest  of  $  2460  for  3  mo.  18  d.  ? 

Solution. 

Time,  108  d.         $  2460  =  Principal. 

Int.  for  60  ^^         =$  24.60;  or  .01  of  principal. 
"      30  "         =     12.30,  or  i  of  60  days'  int. 
"      15  "         =       6.15,  or  I  of  30     ''       " 
"  __§_!!__  =       1.23,  or  tV  of  30  "        " 
"  3  mo.  18  d.  =  $  44.28,  or  the  sum  of  the  ahove. 

51.  What  is  the  interest  of  $480  for  84  days  at  7 J  %  ? 

Solution, 
Time,     84  days  $  480  =  Principal. 

Int.  for  60     "  =  $4.80,  or  .01  of  principal. 

"       20     "  =     1.60,  or  J  of  60  days'  int. 

"      _4     "  =:       .32,  or  i  of  20     '' 

"       84  days  at  6  %    =  $  6.72,  or  the  sum  of  the  above. 
Int.  at  1J%  =      1.68,  or  J  of  6 %  int. 
Int.  at  7^  %  =  1^40,  or  $  6.72  +  $  1.68, 


188  INTEREST. 

52.  What  is  the  interest  of  %  66.42  for  4  mo.  12  d.  at  5  %  ? 

53.  What  is  the  interest  of  $  8000  for  3  mo.  3  d.  at  7  %  ? 

54.  What  is  the  interest  of  1 130.50  for  45  days  at  8  %  ? 

55.  What  is  the  interest  of  $  7500  for  2  mo.  21  d.  at  6  %  ? 

56.  What  is  the  interest  of  1 225  from  Feb.  3,  1880,  to 
May  9,  1881,  at  6  %  ? 

57.  What  is  the  amount  of  $  163.20  from  Dec.  12,  1881,  to 
March  27,  1882,  at  10  %  ? 

58.  What  is  the  amount  of  $  900.65  from  Sept.  16,  1881, 
to  Nov.  8,  1882  ? 

59.  What  is  the  interest  of  $  4000  from  May  12,  1879,  to 
June  24,  1880,  at  4i-  %  ? 

60.  What  is  the  amount  of  $653.63  from  Feb.  11,  1880,  to 
Nov.  9,  1882,  at  7  %  ? 

61.  A  bill  of  goods  amounting  to  $  4498.25  was  paid  at  the 
end  of  60  days,  with  interest  at  5  %.     What  was  the  amount  ? 


^  By  any  of  the  preced; 

Lng  methods  find  the  interest 

Of 

For 

At 

62.   $248 

6  mo.  18  d. 

3|%. 

63.   1845 

13  y.  2  mo.  13  d. 

4%. 

64.   $245.80   * 

2  y.  5  mo.  7  d. 

41%. 

65.   $960 

3  y.  7  mo.  9  d. 

5%. 

66.   $849.50 

8  y.  4  mo.  12  d. 

6%. 

67.   $2846 

3  y.  5  mo.  10  d. 

6|%. 

68.    $180 

1  y.  9  mo.  15  d. 

7%. 

69.   $948.39 

3  y.  11  mo.  6  d. 

n%- 

70.   $862 

4  y.  7  mo.  22  d. 

8%. 

71.   $1500 

1  y.  3  mo.  27  d. 

9%. 

72.   $8400 

2  mo.  17  d. 

7.3%. 

73.   $9398 

1  mo.  18  d. 

10%. 

74.   $479.85 

106  days 

5%. 

75.   $948.25 

89  days 

4i% 

76.   $84.32 

45  days 

4%. 

77.  *  961.18 

111  days 

6%. 

INTEKEST. 

1 

Find  the  amouni 

Of 

From 

To 

At 

78. 

$549.82 

Dec.  14,  '80 

May  5,  '81 

9%. 

79. 

$856.84 

Aug.  17,  '81 

Apr.  4,  '82 

8%. 

80. 

$1248 

Jan.  24,  '81 

Mar.  31,  '81 

6%. 

81. 

$  960.50 

Mar.  5,  '81 

Sept.  9,  '82 

H  %. 

82. 

$  849.25 

May  5,  '81 

Aug.  11,  '81 

5%. 

83. 

$  562.15 

Aug.  15,  '80 

Dec.  29,  '82 

7%. 

84. 

$476.84 

Sept.  30,  '81 

May  6,  '82 

7i  %. 

85. 

$942 

Aug.  31,  '81 

Dec.  30,  '81 

3^^  %. 

86. 

$1728 

Jan.  16,  '80 

Oct.  11,  '81 

8%. 

87. 

$  945.96 

June  4,  '81 

Sept.  10,  '81 

9%. 

88. 

$200 

May  9,  '82 

Aug.  5,  '82 

10%. 

89. 

$  816.42  ^ 

Nov.  4,  '80 

May  1,  '82 

7%. 

90. 

$945.55 

May  19,  '80 

Oct.  19,  '84 

7%. 

91. 

$  624.87 

Sept.  4,  '81 

Dec.  15,  '81 

5%. 

189 


EXACT  INTEREST. 

265.  In  the  computation  of  Uxad  Interest,  as  by  the 
United  States  on  its  securities,  for  parts  of  a  year,  the  actual 
number  of  days  in  each  calendar  month  included  in  the 
time  is  counted,  and  each  day's  interest  made  a  365th  of 
a  year's  interest.     That  is,  to  compute  exact  interest. 

Multiply  the  interest  of  the  jprincijpal  for  one  year  at  the 
given  rate  hy  the  exact  number  of  days  in  the  time,  and  di- 
vide hy  365. 

92.  What  is  the  interest  on  a  United  States  Treasury  cer- 
tificate for  $500  from  April  1,  1881,  to  July  15,  1881,  at  4%? 

$  500  X  .04  =  $  20  Solution.  —  The   exact  time  from 

2  1  April  1, 1881,  to  July  15,  1881,  is  105 

^  — 5^#^=^  =  $5.75  days.     The   interest  of  $500  for  1 

"^  3  year  at  4  %  is  $  20  ;  and  the  interest 

for  105  days  must  be  ^  of  $  20,  which  is  $  5.75f|,  or,  to  the  nearest 

cent,  $5.75. 


190  INTEREST. 

93.   What  is  tlie  exact  interest  on  a  note  for  $  3000  from 
Feb.  15,  1880,  to  June  5,  1880,  at  5%  ? 
**    94.   What  is  the  exact  interest  on  a  $  1000  bond  from  Nov. 
1,  1881,  to  March  15,  1882,  at  4i%  ? 

95.  A  note  for  $  225.50  was  given  March  16,  1881,  bearing 
exact  interest  from  date  at  6  %.  What  sum  should  discharge 
the  note  Jan.  13,  1882  ? 

PROBLEMS  IN  INTEREST. 
Principal,  Interest,  and  Time  given,  to  find  the  Rate. 

96.  At  what  rate  must  $  450  be  on  interest  to  yield  $  81  in 
3  years  ? 

$  450  X  .01  X  3  =  $  13.50  Solution.  —  As  the  interest  of 

1 81  —  $  13  50  =  6  ^  ^^^  ^°^  ^  ^^^^^  ^^  1  %  is  ^  13.50, 

$81  is  the  interest  at  as  many 

per  cent  as  there  are  times  $13.50  in  $81,  or  6. 

97.  The  interest  of  1250  for  1  year  3  months  is  $28.12^. 
What  is  the  rate  per  cent  ? 

266.     Rule  to  find  the  Rate  of  Interest. 

Divide  the  given  interest  hy  the  interest  of  the  principal 
for  the  given  time  at  1  per  cent,  and  the  quotient  will  he  the 
rate. 

Representing  the  principal  by  ^,  the  interest  by  t,  the  time  by  i, 
and  the  rate  by  r,  as  in  Art.  260,  we  ha\'e 

EORMULA.     r  =  t  -^  (^  X  0- 

98.  If  $  1400  yields  $  126  in  1  year  6  months,  what  is  the 
rate? 

99.  At  what  rate  must  $  1000  be  on  interest  to  yield  $  282 
in  4  years  8  monthe  12  days  ? 


INTEREST.  191 

100.  At  what  rate  must  $  416  be  on  interest  to  yield  $  88.64 
in  3  years  16  days  ? 

101.  At  what  rate   must    $  1600  be  on  interest  to  yield 
$46.20  in  66  days? 

102.  At  what  rate  will  $241.20    amount   to    $260.58  m 
6  months  20  days  ? 

103.  At  what  rate  will  $  480  yield  $  52.20  in  2  y.  5  mo.  ? 

104.  What  is  the  rate  of  interest  if  $  640  gains  $  10.56  from 
August  12  to  October  18  ? 

105.  In  1  y.  3  mo.  15  d.  $  960  amounts  to  $  1084.    What  is 
the  rate  ? 

106.  At  what  rate  will  $  444  gain  $  156.695  in  6  y.  5  mo.  ? 

107.  At  what  rate  must  I  invest  a  trust  fund  of  $  25000  to 
secure  a  semi-annual  income  of  $  500  ? 

Principal,  Interest,  and  Rate  given,  to  find  the  Time. 

108.  In  what  time  will  $  450  yield  $  94.50  at  6  %  ? 

Solution.  —  As  1 27  is  the  interest 

$  450  X  .06    =  $  27  of  $  450  for  1  year  at  6  %,  $  94.50  is 

$  94.50  -f-  $  27  =  3 J-  the  interest  for  as  many  years  as 

31  y.  =  3  y.  6  mo.  t^^r®  ^^^  times  $  27  in  $  94.50,  or, 

3-J  years,  equal  to  3  years  6  months. 

109.  The  interest  of  $  140  is  $  49  at  7  %.     How  long  has  it 
been  on  interest  ? 

267.     Rule  to  find  the  Time. 

Divide  the  given  interest  by  the  interest  of  the  jprinciiml  for 
one  year,  and  the  quotient  will  he  the  time. 

Formula,     t  ^  i  -^  (p  X  t). 

110.  How  long  must  $  98  be  on  interest  to  gain  $  23.48  at 
8%? 

111.  How  long  must  $  75  be  on  interest  to  gain  $  6.25  at 

6%? 


192  INTEREST. 

112.   How  long  must  $  3600  be  on  interest  to  gain  $  46.20 
at7%? 
•*r^   113.   How  many  days  must  $  875  bear  interest  to  gain  $  7 
at6%? 

114.  How  long  must  1 9080  be  on  interest  to  gain  $  794.50 
at3j%? 

115.  In  wliat  time  will  $  750,  on  interest  at  6  %  gain  %  750^ 
or  double  itself  ? 

^    116.   I  deposited  $  540  in  a  bank  paying  4  %  simple  interest 
until  it  amounted  to  %  700.     How  long  did  it  remain  ? 

117.  Principal,  %  892  ;  rate,  10  % ;  interest,  %  187.    Eequired 
the  time. 

118.  In  what  time  will  $  12000  yield  $  2500  at  4^  %  ? 

119.  In  what  time  will  a  trust  fund  of  %  4500  amount  to 
$6000  at  31  %? 

'    Interest,  or  Amount,  Time,  and  Rate  given,  to  find  the  Principal. 

120.  What  principal  at  6%  will  gain  $94.50  in  3  years 
6  months  ? 

$  1  X  -06  X  3i  =  $  0.21  Solution.  —  As  the  interest  of 

$  94.50  -^  $  0.21  =  450  $  1  at  6  %  for  3  years  6  months 

is  $  0.21,  $  94.50  must  be  the  in- 
terest of  as  many  dollars  as  $  0.21  is  contained  times  in  $  94.50,  or 
$450. 

121.  What  sum  of  money  at  7%  interest  will  amount  to 
$  320  in  4  years  ? 

Solution.  —  As  the  amount  of 

$  1  X  .07  X  4  =  $  0.28  $  1  at  7  %  for  4  years  is  $  1.28, 

$  1  +  $  0.28  =  $  1.28  $  320  must  be  the  amount  of  as 

$320  -^  $  1.28  =  250  many  dollars  as  $320  is  times 

$  1.28,  or  $  250. 

122.  What  principal  on  interest  at  6  %  will  gain  $  6.25  in 
1  year  4  montlis  20  days  ? 


INTEKEST.  193 

268.     Rule  to  find  the  Principal. 

If  the  interest  is  given,  divide  it  by  the  interest  0/  $  1  at  the 
given  rate  for  the  given  time,  and  the  quotient  will  be  the 
'principal. 
Or, 

If  the  am^ount  is  given,  divide  it  by  the  amount  oj  %1  at  the 
given  rate  for  the  given  time,  and  the  quotient  will  be  the 
'principal. 

Formulas,    p  =  i  -^  (r  x  t)  -,  p  =  a  -^  (1  -\-  r  X  t). 

123.  What  principal  at  7  %  interest  will  gain  $  46.20  in 
66  days  ? 

124.  What  sum  at  4  %  interest  will  amount  to  $318  in 
1  year  6  months  ? 

125.  What  sum  at  5  %  interest  will  amount  to  $  734.20  in 
5  months  10  days  ? 

126.  What  principal  at  6  %  interest  will  give  a  quarterly  in- 
come of  $  210  ? 

127.  How  large  a  sum  in  the  savings-bank  at  5  %  interest 
will  give  a  3^early  income  of  $  1200  ? 

128.  The  interest  on  a  note  for  2  y.  6  mo.  at  7%  was  $  118.23. 
What  was  the  face  of  the  note  ? 

129.  What  sum  must  be  invested  in  stock  paying  3|  %  semi- 
annually to  yield  $  924  per  year  ? 

130.  How  much  must  I  invest  in  U.  S.  4  %  bonds  to  pay  thr 
college  expenses  of  my  son,  $  560  per  year,  with  the  income  ? 

131.  A  gentleman  owns  stock  in  a  manufactory  which  pays 
annually  9  %.  He  receives  quarterly  $  324.  What  sum  has 
he  invested  ? 

132.  Mr.  y.  is  said  to  have  an  income  of  $  5400  per  day. 
If  his  income  from  railroad  stock,  paying  8  %,  is  equal  to  his 
income  from  government  securities,  paying  4%,  what  is  he 
worth  ? 


194  INTEREST. 

PARTIAL  PAYMENTS. 

2oB.      A  Promissory  Note. 
C/ne  yecil  a-J^e^  k/ci^  QJ^ ^laTTztae  ^  ^lay  (344^^'^  o/otane, 

270.  A  Promissory  Note  is  a  written  promise  to  pay 
absolutely  a  specified  sum  of  money  for  value  received. 

271.  The  Promisor,  or  Maker,  of  the  note  is  the  person 
who  makes  the  promise  to  pay. 

272.  The  Promisee,  or  Payee,  is  the  person  to  whom  the 
maker  of  the  note  promises  to  pay  the  money. 

273.  The  Face  of  a  note  is  the  sum  named  in  it. 

274.  A  Negotiable  Note  is  one  payable  to  the  bearer,  or 
to  the  payee's  order. 

275.  The  Indorser  of  a  note  is  the  person  who  WTites 
his  name  on  its  back  as  security  for  the  payment  of  the 
note. 

276.  The  Holder  of  a  note  is  the  person  who  owns  it. 

277.  Partial  Pajrments  are  payments  in  part  of  a  note 
or  debt. 

278.  Indorsements  are  records  of  the  partial  payments 
with  their  dates  made  on  the  back  of  the  note. 

279.  A  note  matwres,  or  is  legally  payable,  on  the  third 
day  after  the  time  named  in  the  note  has  expired. 


INTEREST.  195 

2B0.  A  note  draws  interest  from  maturity  at  the  legal 
rate,  unless  it  contains  the  words  "with  interest."  In 
such  case  interest  accrues  from  the  date  of  the  note. 
Thus, 

The  preceding  note  draws  6  %  interest  from  Dec.  14 
1882,  and  the  following  note  draws  7  %  interest  from  Dea 
28,  1881. 

281.      Form  of  a  Demand  Note. 

/^^^^.  Q^/^?iy,   Mec.   ^§.   ^§§f. 

^a^c  iececzf-ec/.  Q^.    Q/^ci-n/y  ^   ^a. 

282.  The  Supreme  Court  of  the  United  States  has 
adopted  for  finding  the  amount  due  on  a  note  on  which 
partial  payments  have  been  made  the  following,  called 

The  United  States  Rule. 

Find  the  amount  of  the  principal  to  the  tivie  when  the  pay^ 
ment,  or  the  sum  of  the  payments,  equals  or  exceeds  the  inter- 
est  dice.  Then  subtract  such  payment  or  payments  from  the 
amount,  and,  with  the  remainder  as  a  new  principal,  proceed 
as  before  to  the  time  of  settlement. 

133. 

i  304y%%.  Providence,  Dec.  8,  1876. 

»0n  demand,  I  promise  to  pay  J.  B.  Anthony,  or  order,  three 
hundred  four  y^^^  dollars,  with  interest  at  6  %.    Value  received 

William  C.  Thomas. 

Indorsements  :  Sept.  2^,  1877,  $  60  ;  July  4, 1878,  $  90  ; 
Aug.  1, 1879,  $  10 ;  Dec.  2  1879,  $  100.  What  was  due  Jan.  7, 
1881  ? 


196  INTEREST. 

Solution. 

Principal $304.84 

Int.  from  Dec.  S,  1876,  to  SepL.  25,  1877,  9  mo. 

17  d • 14.58 

Amount *  319.42 

1st  payment 60.00 

New  principal %  259.42 

Int.  from  Sept.  25,  1877,  to  July  4,  1878,  9  mo. 

9  d 12.06 

Amount $271.48 

2d  payment 90.00 

New  principal    .- $181.48 

Int.  from  July  4,  1878,  to  Aug.  1,  1879,  12  mo. 

28  d 11.74 

Int.  from  Aug.  1,  1879,  to  Dec.  2, 1879,  4  mo.  1  d.  3.66 

Amount $  196.88 

3d  payment,  less  than  int.  due $  10 

4tli  payment 100 

110.00 

New  principal $  86.88 

Int.  from  Dec.  2,  1879,  to  Jan.  7,  1881,  13  mo. 

5d 5.72 

Amount  due  Jan.  7,  1881 $  92.60 

134. 

$  600.  Springfield,  Jan.  6,  1880. 

For  value  received,  I  promise  to  pay  James  Dennis  &  Co., 
or  order,  six  hundred  dollars,  on  demand,  with  interest  at  7  pei 
cent.  Benjamin  Pool,  Jr. 

Indorsements  :  April  6, 1 880,  $  50 ;  Nov.  21, 1880,  $  60.50; 
March  31,  1881,  $  150.     What  was  due  June  30,  1881  ? 


INTEREST.  i97 

135.  A  note  for  $  750,  dated  Oct.  12, 1880,  had  two  indorse- 
ments,  — Dec.  27,  1880,  $325;  Aug.  7,  1881,  $25.  What 
was  due  July  1,  1882,  at  6  %  ? 

136. 

$  1500.  New  Orleans,  March  10,  1880;_^,^ 

Six  months  after  date,  we  jointly  and  severally  promise  to 
pay  John  Hyde  fifteen  hundred  dollars,  with  interest  at  5  per 
cent.     Value  received.  Joseph  Eaymond. 

Louis  Bernardin. 

Indorsements  :  ISTov.  25,  1880,  $  45  ;  July  20, 1881,  $  500  ; 
Jan.  30,  1882,  $  600.     What  was  due  May  15,  1882  ? 

137.  May  16,  1881,  I  gave  my  note,  on  demand,  with  inter- 
est at  7  %,  for  $  563.50 ;  Sept.  26,  1881,  I  paid  $  250.  What 
was  due  May  16,  1882  ? 

138.  You  borrow,  Feb.  9,  1880,  of  Charles  E.  Lowe,  $  3000 
by  note,  with  interest  at  5  %,  and  pay  $  1000  March  9,  1881, 
and  $  800  Kov.  24,  1881 ;  write  the  note  and  the  indorse- 
ments on  it  in  proper  form,  and  find  the  balance  due  Mr.  Lowe 
Jan.  3,  1882. 

139.  On  a  note  for  $  1200,  dated  Aug.  7,  1880,  drawing 
4  %  interest,  there  were  paid.  May  13,  1881,  $  300  ;  Kov.  23, 
1882,  $  275.     What  was  due  Jan.  1,  1883  ? 

140.  What  is  due  May  15,  1881,  on  a  $  6000  note  drawing 
71%  interest,  on  which  $400  was  paid  Aug.  11,  1879,  $  700 
Dec.  15,  1880,  the  note  being  dated  Jan.  1,  1879  ? 

141.  A  note  for  $  500,  given  Jan.  1,  1879,  at  10  %  interest, 
has  on  it  two  indorsements  of  $  100  each,  paid  on  the  first  day 
of  each  year.     What  was  due  June  19,  1881  ? 

142.  The  face  of  a  note  is  $  2400 ;  its  date,  Aug.  12,  1880 ; 
rate  of  interest,  4^  % ;  indorsements :  Sept.  12,  1881,  $  25 ; 
Oct.  12,  1881,  $  700 ;  settled,  Feb.  15,  1882.     What  was  due  ? 

143.  What  sum  will  discharge  a  note  "Nov,  10,  1881,  for 
$  1728  at  9  %,  dated  Nov.  23,  1878,  which  is  indorsed  as  fol- 
lows:  May  15,  1879,  $248;  Aug.  28,  1880,  $301;  May  30, 
1881,  $  300  ? 


198  INTEREST. 

283.  Business  men,  when  settlement  is  made  within  a 
year  after  interest  begins,  often  make  use  of  the  following, 
called 

The  Merchants'  Rule. 

Find  the  amount  of  the  note  or  debt  from  the  time  of  itb 
beginning  to  draw  interest  to  the  time  of  settlement ;  also,  the 
amount  of  each  payment  froin  its  date  to  the  settlement ;  and 
then  subtract  the  sum  of  the  amounts  of  the  payments  from  the 
amount  of  the  note  or  debt. 

144. 

$  850.  Philadelphia,  Jan.  2,  1882. 

For  value  received,  I  promise  to  pay  John  S.  Moreland,  oi 
bearer,  eight  hundred  fifty  dollars,  on  demand,  with  interest 
at  6  per  cent.  Arthur  Ayer. 

Indorsements:  March  18,  1882,  $200;  May  2,  1882, 
$  150 ;  Aug.  18,  1882,  $  300.     What  was  due  Dec.  2,  1882  ? 

Solution, 

Amount  of  $  850  for  11  mo $  896.75 

$  200  for  8  mo.  14  d $208.47 

"  $150  for  7  mo 165.25 

"         $  300  for  3  mo.  14  d 305.20  $668.92 

$227.83 
145. 
$  1164^%.  St.  Louis,  July  6,  1881. 

For  value  received,  I  promise  to  pay  to  the  order  of  Simeon 
H.  Wright  eleven  hundred  sixty-four  ^^  dollars,  with  inter- 
est at  7  %.  Alfred  Shaw. 

Indorsements  :  Sept.  21,  1881,  $  250  ;  Nov.  22,  1881, 
$315;  March  6,  1882,  $100;  May  17,  1882,  $200.  What 
was  due  on  settlement,  July  6,  1882  ? 


INTEREST.  199 

COMPOUND    INTEREST. 

146.  Alfred  Nickerson  deposits  in  a  savings-bank  $600, 
with  the  understanding  that  at  the  end  of  every  6  months  he 
is  to  receive  interest  on  his  deposit  at  the  yearly  rate  of  4  %. 
How  much  interest  is  due  him  at  the  end  of  6  months  ? 

147.  If  he  does  not  choose  to  draw  this  interest,  it  will  be 
placed  to  his  credit  with  his  original  deposit.  Of  what  sum 
should  he  receive  the  interest  for  the  next  6  months  ? 

148.  How  much  interest  will  be  due  him  from  the  bank  for 
the  second  6  months  ? 

149.  Suppose  that  he  allows  all  his  money  to  remain  in  the 
bank,  on  how  much  will  he  receive  interest  for  the  third 
period  of  6  months  ? 

284.  Compoimd  Interest  is  the  interest  on  both  the  prin- 
cipal and  its  unpaid  interest  added  to  it  at  stated  intervals. 

285.  Interest  may  become  due,  and  made  a  part  of  the 
principal,  or  compounded,  according  to  agreement,  at  the 
end  of  each  year,  half-year,  or  quarter,  or  any  other  period 
of  time. 

150.  What  is  the  compound  interest  and  the  amount  of 
$500  for  2  y.  7  mo.  12  d.  at  6%  ? 

Solution, 

Principal  for  1st  year $500.00 

Interest      "  " 30.00 

Principal  for  2d  year $  530.00 

Interest      "         "        31.80 

Principal  for  7  mo.  12  d $  561.80 

Interest      "        "         " .        20.79 

Compound  amount  for  2  y.  7  mo.  12  d.   .     .  $  582.59 

Given  principal 500.00 

Compound  interest  for  2  y.  7  mo.  12  d.    .     .     $  82.59 


200  INTEREST. 

151.  What  is  the  compound  interest  of  $  750  for  4  years 

at5%? 

286.    Ri'le  for  Compound  Interest. 

Find  the  amount  of  the  given  principal  for  the  first  period 
of  time.  Using  this  amount  as  a  principal,  find  its  amount 
for  the  second  period,  and  so  on  for  the  entire  time.  The  last 
amount,  less  the  given  principal,  will  he  the  compound  in- 
terest. 

Note.  —  When  the  interest  is  compounded  half-yearly,  the  rate  must  be 
considered  one  half  the  yearly  rate,  and  when  quarterly,  one  fourth  the  yearly 
rate.     Interest  compounds  annually  if  nothing  is  said  to  the  contrary. 

152.  What  is  the  compound  interest  of  $  600  for  3  years 

6  months  at  5  %  ? 

153.  What  is  the  compound  interest  of  $  320  for  2  y.  9  mo. 
at  7  %  ? 

154.  What  is  the  compound  interest  of  $  500  for  4  y.  4  mo. 
15  d.  at  4  %  ? 

155.  AVhat  is  the  amount  of  $  1000  for  2  years  at  6  %,  com- 
pounded half-yearly  ? 

156.  What  is  the  amount  of  1 1200  for  1  y.  6  mo.  at  4  %, 
compounded  quarterly  ? 

157.  Willard  Aldrich  deposits  $200  in  a  savings-hank  pay- 
ing 4%  interest,  compounding  half-yearly.  He  withdraws  his 
money  after  three  dividends  have  been  declared.  How  much 
has  he  ? 

158.  Charles  Underhill  borrows  $  2000  for  1  y.  8  mo.  24  d., 
paying  simple  interest  at  6%.     He  lends  it  immediately  at 

7  %  compound  interest.     What  does  he  gain  ? 

159.  What  is  the  compound  interest  of  $  300  for  4  y.  8  mo. 
12  d.  at  8  %,  interest  compounding  semi-annually  ? 

287.  The  computation  of  compound  interest  may  be 
abridged  by  means  of  the  following 


INTEREST. 


201 


TABLE 

SHOWING  THE   AMOUNT    OF  $  1    AT    COMPOUND    INTEREST,  FROM    1    TO    20 

YEARS,  AT  1^,  2,  21,  3,  3^,  4,  5,  6,  7,  8,  9,  and  10  per  cent. 


Yrs. 

1^  per  cent. 

2  per  cent. 

2^  per  cent. 

3  per  cent. 

3|  per  cent. 

4  per  cent. 

1 

1.015000 

1.020000 

1.025000 

1.030000 

1.035000 

1.040000 

2 

1.030225 

1.040400 

1.050625 

1.060900 

1.071225 

1.081600 

3 

1.045678 

1.061208 

1.076891 

1.092727 

1.108718 

1.124864 

4 

1.061363 

1.082432 

1.103813 

1.125509 

1.147523 

1.169859 

5 

1.077283 

1.104081 

1.131408 

1.159274 

1.187686 

1.216653 

6 

1.093442 

1.126162 

1.159693 

1.194052 

1.229255 

1.265319 

7 

1.100843 

1.148686 

1.188686 

1.229874 

1.272279 

1.315932 

8 

1.126491 

1.171660 

1.218403 

1.266770 

1.316809 

1.368569 

9 

1.143388 

1.195093 

1.248863 

1.304773 

1.362897 

1.423312 

10 

1.160539 

1.218994 

1.280085 

1.343916 

1.410599 

1.480244 

11 

1.177947 

1.243374 

1.312087 

1.384234 

1.459970 

1.539454 

12 

1  195616 

1.268242 

1.344889 

1.425761 

1.511069 

1.601032 

13 

1.213550 

1.293607 

1.378511 

1.468534 

1.563956 

1.665074 

14 

1.231753 

1.319479 

1.412974 

1.512590 

1.618695 

1.731676 

15 

1.250229 

1.345868 

1.448298 

1.557967 

1.675349 

1.800944 

IQ 

1.268982 

1.372786 

1.484506 

1.604706 

1.733986 

1.872981 

17 

1.288017 

1.400241 

1.521618 

1.652848 

1.794676 

1.947901 

18 

1.307337 

1.428246 

1.559659 

1.702433 

1.857489 

2.025817 

19 

1.326946 

1.456811 

1.598650 

1.753506 

1.922501 

2.106849 

20 

1.346849 

1.485947 

1.638616 

1.806111 

1.989789 

2.191123 

Yrs. 
1 

5  per  cent. 

6  per  cent. 

7  per  cent. 

8  per  cent. 

9  per  cent. 

10  per  sent. 

1.050000 

1.060000 

1.070000 

1.080000 

1.090000 

1.100000 

2 

1.102500 

1.123600 

1.144900 

1.166400 

1.188100 

1.210000 

3 

1.157625 

1.191016 

1.225043 

1.259712 

1.295029 

1.331000 

4 

1.215506 

1.262477 

1.310796 

1.360489 

1.411582 

1.464100 

5 

1.276282 

1.338226 

1.402552 

1.469328 

1.538624 

1.610510 

6 

1.340096 

1.418519 

1.500730 

1.586874 

1.677100 

1.771561 

7 

1.407100 

1.503630 

1.605781 

1.713824 

1.828039 

1.948717 

8 

1.477455 

1.593848 

1.718186 

1.850930 

1.992563 

2.143589 

9 

1.551328 

1.689479 

1.838459 

1.999005 

2.171893 

2.357948 

10 

1.628885 

1.790848 

1.967151 

2.158925 

2.367364 

2.593742 

11 

1.710339 

1.898299 

2.104852 

2.331639 

2.580426 

2.853117 

12 

1.795856 

2.012197 

2.252192 

2.518170 

2.812665 

3.138428 

13 

1.885649 

2.132928 

2.409845 

2.719624 

3.065805 

3.452271 

14 

1.979932 

2.260904 

2.578534 

2.937194 

3.341727 

3.797498 

15 

2.078928 

2.396558 

2.759031 

3.172169 

3.642482 

4.177248 

16 

2.182875 

2.540352 

2.952164 

3.425943 

3.970306 

4.594973 

17 

2.292018 

2.692773 

3.158815 

3.700018 

4.327633 

5.054470 

18 

2.406619 

2.854339 

3.379932 

3.996019 

4.717120 

5.559917 

19 

2.526950 

3.025600 

3.616527 

4.315701 

5.141661 

6.115909 

20 

2.653298 

3.207136 

3.869684 

4.660957 

5.604411 

6.727500 

202  INTEREST. 

Note  1.  — When  the  time  extends  beyond  the  limits  of  the  table,  find  the 
amount  for  a  convenient  length  of  time,  and  use  this  amount  for  a  new 
principal. 

Note  2.  —  If  the  interest  is  compounded  half-yearly,  take  one  half  the  given 
rate  and  twice  the  number  of  years ;  and  if  compounded  quarterly,  take  one 
fourth  the  given  rate  and  four  times  the  number  of  years. 

16a  What  is  the  compound  interest  of  $  400  for  15  years 
6  months  at  6%? 

Solution, 

Amt.  of  II  for  15  y.  at  6%,  from  table      ....  $2.396558 
Amt.  of  $  400  for  15  y.  at  6  %  =  $  2.396558  X  400  .=  $  958.623 

Int.  of  $  958.623  for  6  mo 28.759 

Amount  of  I  400  for  15  y.  6  mo.  at  6  % $  987.382 

Int.  of  $400  for  15  y.  6  mo.  =  $  987.38  -  $400  .     .  $  587.38 

161.  What  is  the  amount  of  $  500  for  20  years  at  7  %  com- 
pound interest  ? 

162.  What  is  the  compound  interest  of  $  120  for  14  years 
at  8  %  compound  interest  ? 

QUESTIONS. 

254.  What  is  interest  ?  255.  The  principal  ?  256.  The  amount  ? 
257.  The  rate? 

258.  What  is  simple  interest  ?  260.  What  is  the  general  method 
of  computing  simple  interest  1  262.  What  is  the  ^ix  per  cent 
method  ? 

264.  How  may  the  six  per  cent  method  be  shortened  ?  265. 
What  is  the  method  for  finding  the  exact  interest  for  parts  of  a  year  1 

266.  How  do  you  find  the  rate  of  interest,  the  principal,  interest, 
and  time  being  given  1 

267.  The  time,  the  principal,  interest,  and  rate  being  given  ? 
268.  The  principal,  the  interest  or  amount,  time,  and  rate  being 
given  ? 

270.  What  is  a  promissory  note  1     273.    The  face   of  a  note  ? 

277.  What  are  partial  payments  ?     Indorsements  ? 

282.    What  is  the  United  States  Rule  for  partial  payments  1 

284.  What  is  compound  interest  ?     286.    What  is  the  process  of 

•omputing  compound  interest  1 


DISCOUNT.  203 


DISCOUNT. 

288.  1.  What  sum  put  at  interest  at  5  %  will  in  5  years 
amount  to  $  40  ? 

Solution,  —  5  years'  interest  is  .25,  or  J  of  the  principal.  The 
amount  is  |  of  the  principal.  As  $  40,  then,  must  be  |  of  the  prin- 
cipal, I  of  the  principal  is  4  X  i  of  $  40,  or  $  32 

2.  What  sum  on  interest  at  6  %  will  become  $  36  in  3J 
years  ? 

3.  When  an  article  whose  list  price  is  $  25  is  sold  for  cash 
at  10  %  off,  how  much  is  the  deduction  ? 

4.  If  I  borrow  $  200  for  4  months,  and  pay  the  interest  at 
6  %  in  advance,  how  much  is  deducted  from  the  debt  for  the 
interest  ?  ^ 

289.  Discount  is  the  sum  deducted  from  a  debt  or  price. 

TRUE  DISCOUNT. 

5.  Fred  Wood  has  this  day  bought  a  horse  of.  me,  agreeing 
to  pay  me  $  525  for  it  in  1  year  without  interest.  If  I  prefer, 
he  will  pay  me  cash,  provided  I  make  him  an  equitable  dis- 
count from  the  price.  I  propose  a  discount  of  $  15.  He  ob- 
jects, saying  that  I  can  then  take  the  proposed  price,  $  510, 
put  it  at  interest  at  the  current  rate,  5  %,  and  at  the  end  of  the 
year  I  shall  have  more  than  the  original  $  525  which  he  agreed 
to  pay.     What  discount  ought  I  in  justice  to  make  ? 

290.  The  Present  Worth  of  a  debt,  payable  at  a  future 
time  without  interest,  is  the  sum  which  will  amount  to 
the  debt  when  it  becomes  due  if  put  at  interest  at  the 
current  rate. 

291.  True  Discount  is  the  difference  between  the  face  of 
a  debt  and  its  present  worth,  and  is  equal  to  the  interest 
of  the  present  worth  of  the  debt  for  the  given  time. 


204  DISCOUNT. 


WRITTEN    EXERCISES. 


6.  What  is  the  present  worth  of  a  debt  of  1 25.44  due  one 
year  hence,  the  current  rate  of  interest  being  6  %  ?  What  is 
the  true  discount  ? 

$1  X  1.06  =  $1.06  Solution.  —  As  the  amount  of 

$  25.44  -^  1.06  =  $  24  $  1  for  1  year  at  6%  is  $  1.06,  the 

$  25.44  —  $24:  =  $  1.44  present  worth  of  $  1.06,  due  1  year 

hence,  is  $  1 ;  and  as  the  present 
worth  of  $  1.06  is  $  1,  the  present  worth  of  $25.44  must  be  as  many 
dollars  as  $1.06  is  contained  times  in  $25.44,  or  $24.  $25.44  — 
$  24  =  $  1.44,  the  true  discount. 

Note.  — The  process  is  the  same  as  finding  the  principal  (Art.  268),  the  debt 
being  the  amount,  the  present  worth  the  2^'^i'^cip(^l,  and  the  true  discount  the 
interest. 

7.  What  is*  the  true  discount  of  $  192  due  4  years  hence, 
money  being  worth  7  %  ? 

292.     Rule  to  find  the  Present  Worth  and  True  Discount. 

Divide  the  given  debt  by  the  amount  of$l  for  the  given 
time  and  rate,  and  the  quotient  will  be  the  present  worth. 

Subtract  the  present  worth  from  the  debt,  and  the  difference 
will  be  the  true  discount. 

8.  What  is  the  present  worth  of  $  3450,  due  in  1  year  6 
months,  without  interest,  the  current  rate  being  7  %  ? 

9.  What  is  the  true  discount  of  $  172.86,  due  in  3  years 
4  months,  money  being  worth  6  %  ? 

10.  What  is  the  present  worth  of  $  360,  due  in  90  days,  the 
current  rate  being  4  %  ? 

11.  What  is  the  difference  between  the  true  discount  and 
the  interest  on  %  5000  for  2  years  6  months  at  7  %  ? 

12.  What  must  be  the  face  of  a  note,  due  in  2  years  7  months 
15  days,  with  interest,  to  exactly  cancel  a  debt  of  $  347.25, 
due  in  the  same  time,  without  interest,  money  being  worth 
6%? 


X 


DISCOUNT.  205 

COMMERCIAL  DISCOUNT. 

293.  Commercial  Discount  is  a  certain  percentage  de- 
ducted from  the  price  of  an  article,  or  from  the  face  of  a 
bill,  for  cash  payment,  without  regard  to  time. 

294.  The  Net  Price  of  an  article  is  the  selling,  or  list 
price,  less  the  discount. 

13.  When  goods  whose  list  price  is  $  125  are  sold  at  5  %  oft, 
what  is  the  commercial  discount  ? 

14.  What  is  the  net  cash  price  of  a  carriage  hilled  at  $  350, 
on  30  days,  or  6  %  off  for  cash  ? 

15.  What  is  the  discount  on  a  hill  of  goods  invoiced  at 
$  1344.50,  sold  on  30  days,  at  2  %  off  for  cash  ? 

16.  What  is  the  net  cash  value  of  hooks  amounting  as  per 
hill  to  $  460.50,  less  a  discount  of  10  %  and  5%  off  for  cash  ? 

17.  Paid  $  433  for  goods  after  a  discount  of  6  %  had  been 
made  from  the  list  price.     What  was  the  list  price  ? 

18.  What  is  the  difference  in  net  cash  value  between  a  bill 
of  $  1600,  less  a  discount  of  25%  and  5%  off  the  remainder, 
and  the  bill  less  a  discount  of  30  %  ? 

19.  Find  the  amount  of  the  following  bill : 

Boston,  Oct.  19,  1884. 
Taylor  &  Ames, 

Bought  op  Leach,  Shewell,  &  Sanborn. 

200  Greenleaf's  Algebra 95  f      ...  10%  off. 

50  First  Lessons  in  Numbers    ,     .     .  ISJ;^^    ...  7^%  off. 

3  Cases  School  Slates $6.35  ...  20%  off. 

25  Webster's  School  Dictionary     .     .      1.12  .  .  .  12j%  off. 

A  discount  of  2i  %  was  allowed  for  cash  payment. 

20.  Bought  of  A.  L.  Davenport  62  yards  Brussels  carpeting 
at  $1.87J;  118^  yards  3-ply  carpet  at  90/  ;  1  set  parlor  fur- 
niture, $  285 ;  2  sets  black  walnut  chamber  furniture  at  $  125 
and  $  140.  Eeceived  a  discount  of  5  7o.  What  was  the 
amount  of  my  bill  ? 


206  DISCOUNT. 

BANK    DISCOUNT. 

Form  of  a  Discountable  Note. 

/^^4^.  &iim   <foid, /^/y  :ro,  :f§§S. 

(2fcx/y  cuiyd^  a^€e^  <zci^,    Q/  ^lo??z(^e  ^a  ^iciy  ^a  ^t/ie  o^^^ 

295.  The  above  note  may  be  transferred  by  the  in- 
dorsement of  the  payee  (Arts.  272  and  275)  to  any  person 
to  whom  he  may  order  it  paid ;  or  if  the  payee  writes  only 
his  signature  on  the  back,  the  note  becomes  payable  to, 
and  may  be  collected  by,  the  bearer  or  holder. 

Should  the  holder  of  this  note  wish  to  turn  it  into 
cash,  he  can  take  it  at  any  time  before  its  maturity  to  a 
bank.  If  the  bank  officers  are  satisfied  that  the  maker 
and  the  indorsers  are  responsible  parties,  they  will  pay 
to  the  holder  the  face  of  the  note  less  the  interest  on  it 
from  the  day  of  discount  to  the  day  of  maturity,  retain- 
ing the  note  as  security  for  the  money  advanced.  On  the 
day  the  note  matures,  the  maker  of  the  note,  who  has  been 
previously  notified  that  the  note  is  held  by  the  bank,  ap- 
pears at  the  bank  and  pays  the  note. 

296.  Bank  Discount  is  an  allowance  made  to  a  bank  by 
the  holder  of  a  note  for  its  payment  before  maturity. 

It  is  the  simple  interest  of  the  face  of  the  note  from  the 
day  of  discount  to  the  day  of  maturity. 

297.  The  Proceeds  of  a  note  discounted  are  its  face  less 
the  discount. 


DISCOUNT.  207 

298.  Three  Days  of  Grace  are  generally  allowed  by  law 
after  a  note  is  nominally  due,  —  that  is,  after  the  expiration 
of  the  time  named  in  the  note  for  its  payment,  —  before  it 
is  legally  due. 

Note  1.  —  A  note  matures,  or  becomes  legally  due,  on  the  last  day  of  grace, 
or  on  the  day  before  when  the  last  day  of  grace  is  Sunday  or  a  legal  holiday. 

Note  2.  —  The  time  when  a  note  is  due  is  usually  indicated  by  writing  the 
date  when  nominally  due  and  the  date  of  maturity  with  a  line  between.  Thus, 
Aug.  8/ii^  1882. 

299.  The  Term  of  Discount  is  the  time  from  the  day  of 
discount  to  the  day  of  maturity. 

NoteI.  —  When  a  note  is  discounted  at  date,  the  term  of  discount  is  the  time 
specified  in  the  note  plus  three  days  of  grace.  When  the  note  is  discounted 
after  date  it  is  generally  customary  to  use  as  the  term  of  discount  the  exact 
number  of  days  from  the  day  of  discount  to  the  day  of  maturity. 

A  note  is  understood  to  be  discounted  at  date  if  nothing  is  said  to  the  con- 
trary. 

Note  2.  —  When  a  note  is  payable  in  a  certain  number  of  months,  calendar 
months  are  understood,  and  the  note  is  nominally  due  on  the  corresponding 
day  of  the  month,  or  on  the  last  day  of  the  month  when  there  is  no  correspond- 
ing day.     Thus, 

A  3-mo.  note  dated  Dec.  31, 1881,  is  due  March  Sl/^pj-^j  3  1882,  but  a  2-mo. 
note  of  the  same  date  is  due  Feb.  ^^I^^xqIi  3   1882. 

300.     Rule  for  finding  the  Proceeds  of  a  Note. 

Find  the  hank  discount  by  computing  the  simple  interest  of 
the  face  of  the  note  from  the  day  of  discou7it  to  the  day  of 
maturity. 

The  bank  discount  subtracted  from  the  face  of  the  note  gives 
the  proceeds. 

Note.  —  When  an  interest-bearing  note  is  discounted,  the  discount,  must  be 
computed  on  the  amount  due  at  maturity. 

WRITTEN    EXERCISES. 

21.  What  are  the  proceeds  of  a  QO-day  note  for  $  842  dis« 
counted  at  date  at  6  %  ? 


208  DISCOUNT. 

Sohvtion, 

Term  of  discount  93  days.         Face  of  note  $  842.00 

Interest  for       60     " $8.42 

"                 30     " 4.21 

«  3     " .42 

Bank  discount  $  13.05 

Proceeds  $  828.95 

22. 

$  960.  Boston,  Oct.  19,  1881. 

Four  months  after  date,  I  promise  to  pay  to  the  order  of 
George  Horton  nine  hundred  sixty  dollars  at  the  People's 
National  Bank.     Value  received.  Joseph  E.  Libbey. 

The  above  note  was  discounted  Nov.  12,  1881,  at  9  %.  Re- 
quired the  proceeds. 

Solution. 

4  mo.  3  d.  after  Oct.  19,  1881  =.  Feb.  22,  1882,  day  of  ma- 
turity. 

From  Nov.  12,  1881,  to  Feb.  22,  1882  =  102  d.,  term  of  dis- 
count. 

Interest  of  $  960  for  102  d.  at  9  %  =  $  24.48,  bank  discount. 

$  960  -  1 24.48  =  $  935.52,  proceeds  required. 

Find  the  bank  discount  and  proceeds 

23.  Of  a  4-month  note  for  $  875  at  6  %. 

24.  Of  a  6-month  note  for  $  85.60  at  7  %. 

25.  Of  a  90-day  note  for  $  600  dated  Jan.  4,  1881,  and  dis- 
counted Feb.  3,  1881,  at  6%. 

26.  Required  the  proceeds  of  a  4-month  note  for  $  5000  dis- 
counted 1 5  days  after  date  at  8  %. 

27.  What  are  the  proceeds  of  a  6-month  note  for  $  10500, 
bearing  interest  at  6%,  if  discounted  60  days  after  date  at 
5%? 


DISCOUNT.  209 

Find  the  proceeds  of  notes,  using  the  following  condi- 
tions : 


28.  $485.96 

29.  $966 

30.  $1024 

31.  $287 

32.  $648.50 

33.  $984 

34.  $328 
r^35.  $696 

36.  $842.50 

37.  $500 

38.  $8643 

39.  $242.16 

40.  $800 

41.  $560 

42.  $576 

43.  $898.96 


To  find  the  Face  of  a  Note  to  yield  given  Proceeds. 

44.  For  what  sum  must  a  90-day  note  be  written  that  the 
proceeds  may  he  $  500,  discounted  at  6  %  ? 

Bank  discount  of  a  $  1  note  for  93  days  =  $  0.0155 
Proceeds  of  a  $  1  note  =  $  1  —  $  0.0155,  or  $  0.9845 
Face  of  note  required  =  $  500  -^  $  0.9845,  or  $  507.87+ 

Solution.  — As  the  proceeds  of  a  (>1  note  for  93  days  are  $0.9845, 
to  give  $  500  proceeds  the  face  of  the  note  must  be  as  many  dollars 
as  ^500  is  times  $0.9845,  or  $507.87+. 

45.  What  must  be  the  face  of  a  60-day  note  which,  dis- 
counted at  7  %,  will  give  as  proceeds  $  1500  ? 

46.  The  proceeds  of  a  4-month  note,  discounted  at  6  %,.  are 
$  293.85.     What  was  its  face  ? 


Date 

Time  to  run. 

Day  of  Discount. 

Rate. 

Jan.  14 

4  mo. 

Feb.  27 

6% 

Feb.  12 

3  mo. 

Mar.  8 

7% 

Apr.l 

2  mo. 

Apr.  15 

n% 

May  5 

60  d. 

May  27 

8% 

July  10 

90  d. 

Date 

5% 

June  15 

5  mo. 

Aug.  4 

9% 

Sept.  20 

6  mo. 

Nov.  27 

4% 

Aug.  25 

30  d. 

Sept.  1 

4i% 

Oct.  31 

1  mo. 

Nov.  1 

3i% 

Sept.  7 

3  mo. 

Oct.  10 

6% 

Nov.  12 

75  d. 

Dec.  15 

7% 

Jan.  17 

4  mo. 

Feb.  10 

10% 

Dec.  27 

90  d. 

Jan.  15 

3% 

Mar.  5- 

3  mo. 

Mayl 

8% 

Dec.  11 

6  mo. 

Mar.  18 

5% 

Aug.  8 

4  mo. 

Oct.  27 

9% 

210  DISCOUNT. 

301.     Rule  for  finding  the  Face  of  a  Note. 

Divide  the  given  proceeds  by  the  proceeds  of  $  1  for  the  given 
rate  and  term  of  discount, 

47.  The  proceeds  of  a  60-day  note,  discounted  at  7  %,  are 
$  444.48|.     What  was  its  face  ? 

48.  A  merchant  discounted  a  bill  payable  in  6  months,  by 
deducting  the  interest  for  the  time  without  grace  at  10  %,  and 
received  as  the  cash  proceeds  $  1520.  What  was  the  face  of 
the  bill  ? 

49.  Receiving  a  90-day  note,  I  had  it  discounted  at  once  at 
6  %,  and  received  as  proceeds  $  828.95.  What  was  the  face  of 
the  note  ? 

50.  For  what  amount  must  a  note  be  payable  in  8  months, 
so  that  when  discounted  at  1\  %  the  proceeds  may  be  $  483.56  ? 

MISCELLANEOUS     EXERCISES. 

51.  I  owe  a  debt  of  $  924,  payable  without  interest  April  18, 
1882.  What  shall  be  discounted  for  payment  to-day,  Oct.  6, 
1881,  money  being  worth  5  %  ? 

52.  Find  a  year's  interest  of  the  present  worth  of  $  540,  dutf 
12  months  hence  without  interest,  money  being  worth  8  %  ? 

53.  At  what  date  must  a  $  1200  note  have  begun  to  draw 
interest  which  at  6  %  amounted  to  $  1380,  Oct.  15,  1880  ? 

54.  Wood  owes  Davis  $  5811.  He  pays  him  with  a  60-day 
note.  For  what  sum  should  the  note  be  written  to  pay  the 
exact  debt  if  discounted  at  1^?  a  month  ? 

55.  A  4-month  note  is  dated  August  22.  On  exactly  what 
day  must  it  be  paid  to  save  a  protest  ?     Why  ? 

56.  What  is  the  present  worth  of  $477.71,  due  4  years 
hence,  without  interest,  money  being  worth  6  %  ? 

57.  What  is  the  true  discount  on  ^  900,  due  in  72  days,  the 
current  rat*^  being  7  %  ? 


K 


DISCOUNT.  .211 

58.  What  is  the  difference  between  the  interest  and  the 
true  discount  of  $  576^  due  16  months  hence,  at  6  %  ? 

59.  What  are  the  proceeds  of  a  note  for  $  368  payable  in 
90  days,  discounted  at  bank  at  6  %  ? 

60.  On  what  month  and  day  will  a  note  for  60  days,  dated 
Jan.  31,  1882,  become  legally  due  ? 

61.  A  man  was  offered  $  3675  in  cash  for  his  house,  or 
$  4235  in  3  years  without  interest.  He  accepted  the  latter 
offer.  Did  he  gain  or  lose,  and  how  much,  money  being  worth 
7%? 

62.  Wishing  to  borrow  $  500  at  a  bank,  for  what  sum  must 
my  note  be  drawn  at  30  days  to  obtain  that  amount,  discount 
being  6  %  ? 

63.  What  must  be  the  face  of  a  note,  due  in  45  days,  that, 
when  discounted  at  a  bank  charging  7  %  interest,  will  enable 
me  to  take  up  my  note  for  $  750,  that  has  been  on  interest  at 
7^^  %  for  3  months  and  15  days  ? 

64.  Pratt,  Davis,  &  Co.  sold  an  acre  of  land,  which  cost  them 
$  400,  at  5  cents  per  square  foot,  taking  in  payment  a  6  mo.- 
note  which  they  immediately  get  discounted  at  the  Maverick 
Bank,  at  5  %.     What  were  their  profits  ? 

QUESTIONS. 

289.  What  is  discount?  290.  What  is  the  present  worth  of  a  debt) 
291.  What  is  true  discount  ?  292.  How  is  the  present  worth  found  1 
The  true  discount  ? 

293.  What  is  commercial  discount  ?  294.  What  is  the  net  price  of 
an  article  ? 

296.  What  is  bank  discount  ?  297.  What  are  the  proceeds  of  & 
note? 

298.  When  is  a  note  said  to  mature  ?  299.  What  is  the  term  of 
discount  ?  300.  How  are  the  proceeds  of  a  note  found  ?  301.  How 
do  you  find  the  face  of  a  note  to  yield  given  proceeds  ? 

274.  What  is  a  negotiable  note  ?  275.  What  responsibility  does  a 
person  incur  by  indorsing  a  note  ? 


212  STOCK   INVESTMENTS. 


STOCK    INVESTMENTS. 

302.  1.   A  company  start  business  with  $  10000.     Into 
how  many  shares  of  $  100  each  can  this  be  divided  ? 

2.  How  much  do  10  shares  of  $  100  each  represent  ? 

3.  What  is  the  value  of  five  shares  of  $  100  each  at  a  dis- 
30unt  of  20  %  ? 

4.  When  $  100  shares  sell  at  $  120  each,  how  much  is  the 
advance  on  the  original  value  ? 

5.  When  a  $  100  share  sells  at  $  125,  what  per  cent  is  the 
selling  price  above  the  original  value  ? 

6.  If  you  own  10  shares  of  $  100  each,  and  receive  as  the 
profits  $  60,  what  per  cent  are  the  profits  ? 

303.  A  Share  is  one  of  the  equal  parts  into  which  the 
capital  of  a  corporation  is  divided. 

The  share  is  usually  of  the  original  value  of  $  100,  and 
may  be  so  considered  unless  otherwise  denoted. 

304.  Bonds  are  the  interest-bearing  notes  of   govern- 
ments or  corporations. 

The  interest  on  bonds  is  usually  paid  quarterly  or  semi- 
annually. 

A  Coupon  is  the  interest  certificate  attached  to  a  bond. 

305.  Bonds  are  commonly  named   according  to  their 
rate  of  interest  and  date  of  maturity.     Thus, 

U.  S.  4J's  '91,  means  United  States  Bonds  bearing  4^ 
per  cent  interest  and  payable  in  1891. 

306.  Stocks  are  the  shares  of  companies  and  the  bonds 
of  governments  and  corporations. 

307.  The  Par  of  stocks  is  their  face  value.     Thus, 
When  a  stock  is  quoted  at  105,  it  is  worth  105  %of  its 

face  value. 


STOCK    INYESTMENTS.  213 

308.  The  Market  Value  of  stocks  is  the  price  at  which 
they  sell.  Stocks  are  at  a  p''emium  when  the  market  value 
is  above  par,  and  at  a  discount  when  the  market  value  is 
helow  "par. 

309.  A  Stock  Certificate  is  a  document  signed  by  the 
officers  of  a  corporation  specifying  the  number  of  shares 
owned  by  the  holder. 

310.  A  Dividend  is  a  sum  divided  among  the  stock- 
holders as  the  profits  of  the  business. 

311.  An  Assessment  is  a  sum  required  of  the  stock- 
holders to  meet  the  losses  or  expenses  of  the  business. 

312.  The  market  value  of  leading  stocks  and  bonds  at 
commercial  centers  is  given  in  the  daily  papers.  The  fol- 
lowing is  an  extract : 


5  Union  Pacific 59| 

60  N.  Y.  Central 112f 

200  Erie  K.  R 34|^ 

114  Am.  Bell  Telephone  .     .  205 

13  Continental  Mills     ...  90^ 

2  Exchange  Bank     ....  145 

50  Eastern  R.  R 123 


Stock  Quotations. 

U.  S.  4's  reg. 
U.  S.  4's  coup. 


$1000  So.  Kan.  5's  .  .  . 
$5000  Pacific  6's  '95  .  . 
$5000  Ogd's  «&  Lake  C.  6's 
$10000  N.  Y.  and  N.  E'.  6's 
$5000  Wisconsin  Cent.  2ds. 


127 
127 

100 
125 
100 
106J 
57| 


313.  Brokerage  is  computed  on  the  par  value  of  stocks, 
and  the  usual  rate  is  either  |  %  or  ^  %. 

314.  The  rules  of  percentage  and  interest  already  given 
apply  to  stocks,  the  premium,  discount,  dividend,  and  assess- 
ment  heing  always  a  percentage  of  the  par. 

7.  What  must  be  paid  for  %  5000  Union  Pacific  Eailroad 
Bonds  at  114,  brokerage  \%2 

Solution.— 114%  +  1%  ^  1141%,  and  114|%  of  $5000  = 
85712.50,  Ans. 


214  STOCK   INVESTMENTS. 

8.  What  will  20  shares  of  New  Jersey  Central  Eailroad 
cost,  at  102|,  brokerage  i  %  ? 

9.  How  much,  including  brokerage  at  |  %,  must  be  paid  for 
$ 30000  of  U.S.  4's  at  1131? 

10.  When  gold  is  at  lOlJ,  what  is  the  value  in  currency 
Df  S1250  in  gold? 

11.  When  the  cost  of  20  shares  of  the  Atlantic  National 
Bank,  including  I  %  brokerage,  is  %  2200,  what  is  the  market 
value  per  share  ? 

Solution. 

$  100  X  20     =  $  2000,  par  value. 

^  %  of  $  2000  r==  $  5.00,  the  brokerage. 

%  2200  —  $  5  =  $  2195,  the  market  value. 

%  2195  -^  20  =  $  109.75,  the  market  value  of  1  share. 

12.  When  the  cost  of  25  shares  of  the  Adams  Express  Com- 
pany, including  brokerage  at  ^%,  is  $3206.25,  what  is  the 
market  value  per  share  ?  t 

13.  When  gold  is  at  lOlf ,  what  is  the  value  in  gold  of 
%  126.75  in  currency  ? 

14.  How  should  Pacific  Railroad  bonds  be  quoted  when  10 
hundred-dollar  bonds  cost,  including  brokerage  ^  %,  $  1302.50  ? 

15.  What  income  will  be  realized  from  investing  $  1905  in 
6  %  stock  bought  at  95,  allowing  \  %  for  brokerage  ? 

Solution. 

%  1905  -f-  .95i  =  $  2000,  par  value. 

$  2000  X  .06    =  $  120,  yearly  income.     • 

16.  What  will  be  the  income  from  investing  $  2650  in  State 
5's  at  105|,  brokerage  \%? 

17.  How  much  can  be  realized  yearly  from  an  investment 
of  $  6900  in  a  ^  %  stock,  bought  at  S^,  brokerage  i  %  ? 

18.  Which  will  yield  the  greater  income  in  amount,  Citj' 
6^8,  at  1055,  purchased  for  $2650,  or  State  5's  at  104J,  pui> 
chased  for  $  3135,  brokerage  in  both  cases  being  J  %  ? 


STOCK   INVESTMENTS.  215 

19.  How  much  must  be  invested  in  5  %  stock,  purchased  at 
103,  to  afford  an  income  of  $  800  ? 

Solution. 

%  800  ~  .05  =  $  16000,  par  value  of  stock. 
$  16000  X  1.03  =  $  16480,  amount  to  be  invested. 

20.  How  much  must  I  invest  in  Government  4J's  at  105|^ 
to  secure  annually  $  900  ? 

21.  How  much  must  be  invested  in  7%  railroad  bonds  at 
108f ,  brokerage  \  %,  to  afford  an  income  of  $  1050  per  annum  ? 

22.  What  sum  must  be  invested  in  U.  S.  5  %  bonds  of  $  500 
each,  at  108|,  brokerage  \  %,  to  secure  a  yearly  income  of 
$2500? 

23.  When  a  6  %  stock  is  at  95|,  brokerage  \  %,  what  rate 
of  income  on  the  investment  v/ill  the  stock  yield  ? 

Solution.  —  The  annual  income  of  a  share  of  6  %  stock  is  $  6.  If 
the  cost  is  $  96,  the  income  is  -^,  or  y\,  or  .6 J  %,  of  the  cost. 

24.  What  is  the  rate  of  income  on  E-ailroad  5's  at  110,  no 
allowance  for  brokerage  ? 

25.  Which  will  yield  the  greater  per  cent  income,  Railroad 
6's  bought  at  120,  or  5  %  stock  bought  at  105  ?  How  much  ? 

26.  What  per  cent  income  will  land-grant  8's  purchased  at 
125  yield  ? 

27.  How  much  must  be  paid  for  a  5  %  stock  that  the  in- 
vestment shall  yield  6  %  ? 

Solution.  —  A  5  %  stock  yields  f  5  on  an  investment  of  $  100.  But 
if  the  S  5  is  6  %  of  the  investment,  1  %  of  it  is  ^  of  $  5,  and  100  %  of 
the  investment  100  X  i  of  $  5,  or  $  83J. 

28.  How  much  must  I  pay  for  Eailroad  6's,  that  my  invest- 
ment shall  yield  7  %  ? 

29.  At  what  price  must  I  purchase  8  %  stock  that  the  in- 
vestment shall  pay  6  %  ? 


r^ 


216  EXCHANGE. 


EXCHANGE. 


315.  1.  How  can  you  pay  a  creditor  in  ISTew  Orleans  $  800 
without  actually  sending  him  the  money  ? 

2,  What  Vrill  an  order  of  John  Hall  of  Chicago  on  Abram 
Brown  of  New  York  for  $  300  cost,  payable  at  sight,  ii  ^% 
premium  is  charged  for  it  ? 

3.  What  will  an  order  for  $  250,  payable  at  sight,  cost,  if 
purchased  at  |^  %  discount  ? 

316.  A  Draft  is  a  written  order  for  the  payment  of 
money,  made  in  one  place,  and  payable  in  another. 

317.  The  Draivei'  is  the  maker  of  a  draft ;  the  Drawee, 
the  party  to  whom  it  is  addressed ;  and  the  Payee,  the 
party  to  whom  it  is  payable. 

318.  A  Sight  Draft  is  a  draft  payable  on  presentation 
to  the  drawee. 

319.  A  Time  Draft  is  a  draft  payable  at  a  time  named 
after  presentation,  or  after  date. 

320.    Form  of  a  Sight  Draft. 
om^  ^n(^u4ci?ic(  c/otuzid,  anc/  cnai^  ^  ^de  account  oj- 

321.  If  the  drawee  agrees  to  pay  the  money  specified 
in  the  draft,  on  its  presentation  he  writes  his  name  under 
the  word  "  Accepted "  across  its  face.  This  act  is  called 
the  Acceptance  of  the  draft. 


EXCHAITGE.  217 

322.  Exchange  is  the  method  of  making  payments  by- 
means  of  drafts. 

The  exchange  is  at  par  when  a  draft  sells  for  its  face. 
Three  days  of  grace  are  usually  allowed  on  time  drafts. 

DOMESTIC    EXCHANGE. 

323.  Domestic  Exchange  is  between  persons  in  the  same  ' 
country. 

WRITTEN    EXERCISES. 

4.  What  is  the  cost  of  a  sight  draft  for  $452  at  1J% 
discount  ? 

Solution. 

$  452  X  0.015  =  $  6.78,  discount. 

$  452  -  I  6.78  =  $  445.22,  cost  of  the  draft. 

5.  What  is  the  cost  of  a  draft  on  ]^ew  York  for  $  1164  at 
1  %  premium  ? 

6.  How  much  must  be  paid  for  a  draft  on  St.  Louis  for 
$4000  at  2^%  discount? 

7.  What  is  the  cost  of  a  draft  of  the  Girard  National 
Bank,  Philadelphia,  on  the  National  Bank  of  Commerce, 
Boston,  for  $  2517.70  at  \  %  premium  ? 

8.  Find  the  face  of  the  draft  at  1|  %  discount  that  can  be 
bought  for  $  445.22. 

Solution. 

$  1  -  $  0.015  =  $  0.985,  cost  of  $  1  of  draft. 
$  445.22  -f-  0.985  =  $  452,  face  of  the  draft. 

9.  How  large  a  draft  at  ^  %  premium  can  be  bought  for 
$2520.84? 

10.  Bought  a  draft,  at  21%  discount,  for  $3900.  What  was 
its  face  ? 


218  EXCHANGE. 

11.  A  merchant  in  Mobile  bought  a  draft  on  New  York  a* 
1  %  premium  for  $  1175.64.     What  was  the  face  of  the  draft  ? 

12.  A  man  in  Trenton  bought  a  draft  on  Richmond  at  |  % 
discount,  for  $  447.18f .     What  was  the  face  of  the  draft  ? 

13.  What  must  be  paid  for  a  draft  of  $  1000^  on  Hartford, 
at  30  days,  interest  at  6  %,  when  exchange  is  at  2  %  premium  ? 

$  1  X  1.02  =  $  1.02  Solution.  —  The  cost  of  $  1  at 

$  1.02  X  1000  =  $  1020  sight  at  2  %  premium  is  $  1.02, 

$1000  X  0.0055  =  $5.50  ^'^^  ^^  *  ^^^^  i«  $1^20.     As 

$  1020  -  $  5.50  =  $  1014.50    ^^^  ^^''^^^  ^^^  ^^  ^^^^  ^^  ^^^» 

interest  at  the  given   rate   for 

that  time,  amounting  to  $5.50,  must  be  deducted  ;  ana  the  proceeds, 

$  1014.50,  wiU  be  the  cost  of  the  draft. 

14.  What  must  be  paid  for  a  draft  of  $  1500,  at  60  days,  at 
7  7oj  exchange  being  at  J  %  discount  ? 

15.  I  wish  to  obtain  a  draft  on  Boston  for  $  3000,  at  60 
days,  interest  at  6  %,  exchange  being  at  1  %  premium.  What 
must  I  pay  for  it  ? 

^      16.   What  is  the  face  of  a  30  days'  draft,  at  2  %  premium, 
which  can  be  bought  for  $  2029,  interest  at  6  %  ? 

Solution. 

$  1  +  $  0.02  =  $  1.02,  cost  of  $  1  at  sight. 

$  1  X  0.0055  =  $  0.0055,  int.  of  $  1  for  33  days. 
$  1.02  —  $  0.0055  =  $  1.0145,  cost  of  $  1  of  exchange. 
$  2029  -^  1.0145    =  $  2000,  face  of  the  draft. 

17.  How  large  a  draft  on  Philadelphia,  at  par,  at  30  days, 
can  be  bought  for  $  3978,  interest  6  %  ? 

18.  What  is  the  face  of  a  60  days'  draft  at  |  %  discount, 
which  can  be  bought  for  $  491.37i,  money  being  worth  7  %  ? 

19.  How  large  a  30  days'  draft  can  I  buy  for  $  2998.50,  in- 
terest  at  6  %,  and  exchange  at  1  %  premium  ? 


EXCHANGE.  219 

FOREIGN    EXCHANGE. 

324.  Foreign  Exchange  is  between  persons  in  different 
countries. 

In  foreign  exchange  drafts  or  bills  are  expressed  in  the 
money  of  the  country  in  which  they  are  payable. 

325.  English  or  Sterling  Money  is  expressed  in  pounds, 
shillings,  pence,  and  farthings. 

TABLE. 

4  farthings  (far.)  are  1  penny,  d, 
12  pence  "    1  shilling,  s. 

20  shillings  "    1  pound,  £, 

Also, 
10  florins  (fl.)  are  1  pound,  £. 

326.    The  Pound  Sterling 

is  represented  by  a  gold 
coin,  the  Sovereign  (sov.), 
whose  value  is  $4.8665. 

327.  French  Money  is  expressed  in  francs  and  centimes  ; 
a  franc  being  100  centimes. 

A  franc  has  the  value  of  $0,193,  and  about  5.18  francs 
are  equivalent  to  a  dollar. 

328.  The  Money  of  the  German  Empire  is  expressed  in 
raarJcs  (reichmarken)  and  pe7inies  (pfennige) ;  a  mark  being 
100  pennies. 

A  mark  is  equivalent  to  $  0.238,  and  4  marks  are  about 
95  cents. 

Note.  —  For  the  value  of  other  foreign  coins  as  fixed  by  the  U.  S.  govern- 
ment refer  to  the  table  in  the  Appendix. 


220  EXCHANGE. 

329.  sterling  Bills,  or  drafts  on  England,  Ireland,  and 
Scotland,  are  quoted  at  the  exchange  value  of  a  sovereign 
or  pound  sterling  in  United  States  dollars. 

Exchange  on  Paris,  Antwerp,  and  Geneva  is  quoted  at  a 
certain  number  of  francs  per  dollar ;  and  on  Bremen,  Ham- 
burg, Frankfort,  and  Berlin  at  a  certain  number  of  cents 
per  4  reichsmarks.     Thus, 

Foreign  exchange  may  .be  quoted  as  follows :  Sterling 
sight  4.861  @  4.87,  60  days  4.85;  Francs  sight  5.18  @ 
5.18|^,  60  days  5.15^  @  5.15|;  and  Eeichsmarks  sight  94-| 
@  94f,  60  days  93-7  @  941 

330.  Bills  of  exchange,  or  drafts,  on  foreign  countries 
are  usually  made  in  sets  of  three  of  the  same  tenor  and 
date,  named  first,  second,  and  third  of  exchange.  Any  one 
of  the  set  being  paid,  the  others  are  void. 

20.  Find  the  cost  in  Boston  of  the  following  bill  drawn  on 
London,  exchange  at  4.85. 

of  -^^  (U2>??ze  cui^    C177.CO    -^no^    tin/Kzcc/,  /^^y  ^  ^^^  oiaei^  o/^ 
(^nirnuet     ^Jr.      /raMel  one    duncAec/  t^ccc^y  ^lounc/d   ecaM 

(zccouTi^  0/  '^cc^i,   .£^&a^oc/y,  £^  ^o. 

Solution.  —  £  160  8.s.  =  £  1G0.4  ;  $4.85  X  160.4  =  $777.94. 

21.  When  exchange  on  London  is  at  4.85,  what  will  be  the 
face  of  a  draft  that  can  be  bought  for  f  777.94  ? 


EXCHANGE.  221 

22.  Bought  a  set  of  exchange  on  England  for  £  1320  10s, 
at  4.87^.     What  was  the  cost  ? 

23.  rind  the  value  in  New  York  of  a  set  of  exchange  on 
Paris  for  2380  francs  at  5.15. 

24.  Andrew  Taylor,  of  Providence,  wishes  to  remit  1500 
francs  to  Antwerp.  What  will  be  the  cost  of  a  draft  for  that 
sum,  exchange  at  5.19  ? 

25.  When  exchange  on  Paris  is  5.20,  how  many  francs  of 
exchange  will  $  3195  buy  ? 

26.  How  much  must  be  paid  for  a  set  of  exchange  on  Ham- 
burg for  1304  reichsmarks,  exchange  at  95  ? 

27.  When  exchange  on  Berlin  is  95^,  what  must  be  the 
face  of  a  draft  that  $1420.20  will  purchase  ? 

28.  Find  the  cost  of  a  set  of  exchange  on  London  at  60  days 
for  £  1254  15s.  6d.,  exchange  being  quoted  at  4.87J. 

29.  What  will  be  the  face  of  a  draft  in  francs  that  can  be 
bought  for  $  1042.50,  exchange  being  5.21J  ? 

QUESTIONS. 

316.  What  is  a  draft?  317.  Who  is  the  drawer?  The  drawee? 
The  payee  ? 

318.  What  is  a  sight  draft  ?     319.  A  time  draft  ? 

321.  How  is  a  draft  accepted?  322.  How  many  days'  grace  are 
usually  allowed  on  time  drafts  ? 

322.  What  is  exchange  ?     323.  Domestic  exchange  ? 

324.  What  is  foreign  exchange  ?  325.  How  is  English  money  ex- 
pressed ?  Recite  the  table.  326.  What  is  the  value  of  a  pound  or 
sovereign  ? 

327.  How  is  French  money  expressed  ?  What  is  the  value  of  a 
franc  ?  328.  What  is  the  money  of  the  German  Empire  ?  What  is 
the  value  of  a  mark  ? 

329.  How  is  exchange  on  London  quoted  ?  On  Paris,  Antwerp, 
and  Switzerland  ?     On  Bremen,  Hamburg,  Frankfort,  and  Berlin  ? 

330.  How  are  drafts  on  foreign  countries  usually  made  ? 


i- 


222         "  /     •"        AVERAGE   OF   PAYMENTS. 


AVERAGE    OF   PAYMENTS. 

331.  1.  How  long  should  $  1  be  kept  to  equal  the  use  of 
$2  for  1  month?  % 

2.  In  how  many  months  will  the  interest  of  $  6  balance  the 
interest  of  $  18  for  4  months  at  the  same  rate  per  cent  ? 

Solution.  —  $  6  being  but  J-  of  $  18  will  require  three  times  as  many 
months  to  gaui  as  much  interest  as  $  18  at  the  same  rate,  or  3  X  4 
months,  or  12  months. 

3.  The  interest  of  %  15  for  2  months  is  balanced  by  the  in» 
terest  of  $  1  in  how  many  months  ?     Of  ^3?     Of  $5? 

4.  If  I  should  be  allowed  the  use  of  $  50  for  3  months,  how 
long  in  return  should  I  lend  $  2^  ? 

332.  Average,  or  Equation  of  Payments,  is  the  process  of 
finding  when  several  debts,  due  at  different  times,  may  be 
paid  at  one  time  without  loss  to  either  debtor  or  creditor. 

333.  The  Average,  or  Equated  Time,  is  the  date  of  pay- 
ment. 

WRITTEN    EXERCISES. 

5.  July  1,  A  owes  B  $  100 ;  of  which  $  20  is  due  in  2  months, 
$40  in  3  months,  $30  in  4  months,  and  $10  in  5  months. 
When  may  the  $  100  be  equitably  discharged  by  a  single 
payment  ? 

Solution, 

A  is  entitled  to  2  months'  use  of  $  20  =    40  months'  use  of  $  1 
cc  u     3       u  u     $40  =  120        "  '' 

u  u     4       u  u     $30  =  120         "  " 

"  "     5      "  "     $10=    50        "  " 

A  is  entitled  to  330  months'  use  of  $1, 

which  is  equivalent  to  the  use  of  $  100  for  yj^  of  330  mo.,  or 

3^5^  mo.,  or  3  mo.  9  d.,  which,  added  to  July  1,  gives  the 

equated  time,  Oct.  10. 


Or,  briefly, 


AVERAGE   OF   PAYMENTS.  223 

20  X  2  mo.  :=    40  mo. 

40  X  3   "    =  120   " 

30  X  4  "  =  120  " 
_10  X  5  "  =  _50  " 
100  )330   ^^ 


3^^  mo.,  or  3  mo.  9  d. 
July  1  +  3  mo.  9  d.  =  Oct.  10. 

6.  What  is  the  average  time  of  paying  $  200  due  April  1, 
1200  due  May  11,  and  $  400  due  June  30  ? 

Solution. 

April  1,  $  200  is  due. 

May  11,  or  40  days  after  April  1,  $  200  is  due. 

June  30,  or  90  "  "         $400     " 

200 

200  X  40  days  =    8000  days 
400  X  90  days  =  36000     " 
800  )  44000     " 

55  days 
April  1  +  55d.  =  May  26,  the  average  time. 

334.     Rule  to  find  the  Average  Time  of  Payment. 

Multiply  each  debt  by  its  term  of  credit,  and  divide  the  sum 
of  the  products  by  the  sum  of  the  debts.  The  quotient  will  be 
the  average  term,  of  credit.  This  added  to  the  date  from  which 
the  credits  were  reckoned  luill  give  the  average  time  of  pay 
ments. 

Note.  —  When  the  cents  are  50  ar  more,  reckon  them  as  one  dollar ;  but  if 
less  than  50,  disregard  them.  Also,  when  in  any  result  the  fraction  of  a  day  ia 
^  or  more,  reckon  it  one  day;  otherwise,  disregard  it. 


224  AVERAGE   OF   PAYMENTS. 

7.  I  have  purchased  goods  of  A.  B.  Blake,  as  follows  :  Jan- 
uary 3,  a  bill  of  $  150  on  30  days  ;  January  15,  a  bill  of  $  125 
on  3  months  ;  and  February  1,  a  bill  of  $  200  on  60  days. 
Find  the  average  time  of  payment. 

8.  May  7,  A  owes  B  $  100,  of  which  $  40  is  to  be  paid  in  3 
months,  and  $  60  in  5  months.  Find  the  average  time  of  pay 
ment. 

9.  John  Oldham  owes  Henry  Smith  $  1000,  $  250  of  which 
is  due  now,  $  350  in  2  months,  and  the  remainder  in  6  months. 
What  is  the  average  time  of  payment  ? 

10.  May  16,  1881,  Joseph  Milton  owes  $  169.85,  payable  in 
40  days,  $  200.15  in  60  days,  and  $  150  in  90  days.  Find  the 
equated  time. 

11.  Three  bills  are  due  as  follows  :  April  15,  $  200  ;  May  1, 
$  310.50;  June  1,  1 160.25.  What  is  the  average  date  of  pay- 
ment ? 

12.  Find  the  average  time  for  paying  $  800  due  in  30  days, 
$  500  in  60  days,  and  1 120  in  90  days. 

335.  When  there  is  a  common  term  of  credit,  we  may 
find  the  average  time,  withottt  regard  to  that  teriiiy  and  then 
add  it  to  the  result. 

13.  Albert  Thayer  bought  merchandise  as  follows  on  60 
days  :  July  5,  1881,  $  600 ;  July  15,  1 400 ;  August  10,  $  500. 
Find  the  average  time  of  payment. 

14.  Bought  the  following  bills  on  4  months  :     September  9, 

1880,  $  140  ;  October  9,  $  160  ;  November  6,  $  200.     What  \^ 
the  average  time  for  payment  ? 

15.  Three  60-day  notes  bear  date  as   follows  :    April  11, 

1881,  1450;  April  30,  $600;  May  16,  $400.     What  is  the 
average  date  of  maturity  ? 

16.  Bought  goods  on  6  months'  credit  as  follows :  July  2, 
1881,  $225;  August  4,  1360;  September  10,  1500;  Septem« 
ber  24,  1 320.  When  shall  a  note  to  settle  for  the  whole  be 
made  payable  ? 


X 


REVIEW.  225 

REVIEW. 
ORAL    EXERCISES. 

336.  1.  I  have  $  120.  If  I  pay  away  25  %  of  it,  what  sum 
shall  I  have  left  ? 

2  Sold  a  horse  which  cost  me  $  225  at  10%  profit.  What 
did  I  get  for  him  ? 

3.  Bought  a  watch  for  $  75  and  sold  it  for  I  84.  What  was 
the  gain  per  cent  ? 

4.  When  goods  are  sold  for  |  of  their  cost,  what  is  the  gain 
per  cent  ? 

5.  If  J  of  f  of  the  cost  of  my  horse  is  the  cost  of  my  chaise, 
what  per  cent  is  the  cost  ol  the  horse  more  than  the  cost  of 
the  chaise  ? 

6.  A  knife  which  cost  me  31J  cents  was  sold  for  25  cents. 
What  was  the  loss  per  cent  ? 

7.  At  2^  %,  what  is  the  premium  on  an  insurance  of  $  3000  ? 

8.  An  agent  received  $40.50  for  selling  goods  at  5%  com- 
mission.    What  was  the  amount  of  goods  sold  ? 

9.  What  will  be  the  interest  of  $  550  at  6  %  for  2  years 
6  months  ? 

10.  What  principal  at  6  %  interest  in  2  years  6  months  will 
give  I  66  interest  ? 

11.  What  principal  in  5  years  at  5  %  will  amount  to  $  50  ? 

12.  The  interest  of  $  550  for  2  years  6  months  is  $  66.  What 
is  the  rate  per  cent  ? 

13.  What  is  the  market  value  of  6  shares  of  stock  at  112  ? 

14.  What  is  the  cost  of  a  draft  for  $  400  at  2-i  %  premium  ? 

15.  At  what  price  must  a  4  %  stock  be  bought  for  a  5  %  in- 
vestment ? 

16.  How  much  must  be  paid  for  a  10  %  stock  that  8  %  may 
be  realized  on  the  investment  ? 

15 


226  I  REVIEW. 


^ 


WRITT  EN    EXERCISES. 

17.  21  is  what  %  of  3|  ? 

18.  If  35  %  of  my  money  is  $  2359,  what  is  my  money  ? 

19.  Of  goods  worth  $  1200,  one  fourth  is  sold  at  a  profit  <>i 
15  %.    For  how  much  must  the  remainder  be  sold  to  gain  17  %  ? 

20.  Bought  a  horse  for  $  360.  which  was  20  %  less  than  his 
real  value,  and  sold  him  for  30  %  more  than  his  real  value.  Re- 
quired the  selling  price. 

21.  A  class  of  50  pupils  miss  75  words  in  spelling  10  each. 
What  per  cent  of  words  were  correctly  spelled  ? 

22.  What  per  cent  of  the  year  1880  expired  at  midnight, 
June  15  ? 

23.  Bought  a  bill  of  goods  which,  at  5  %  off  for  cush, 
amounted  to  $  232.75.     How  much  was  the  discount  ? 

24.  Bought  12  Webster's  Unabridged  Dictionaries  at  $9.50, 
10%  off,  and  15  Longfellow's  Poems  at  $  1.50,  5  %  off.  Paid 
cash,  and  received  an  additional  discount  of  3  %.  Required  the 
cost. 

25.  At  7J  %  interest,  how  much  is  due,  July  5,  1881,  on  a 
note  for  $325,  dated  Jan.  7,  1880  ? 

26.  How  long  must  $922  be  on  interest  at  5%  to  gain 
$53.78i? 

27.  For  what  sum  must  a  note  be  made  payable  in  60 
days,  so  that  when  discounted  at  6%  the  proceeds  may  be 
$593.70? 

28.  At  what  rate  of  interest  will  $  640  amount  to  $  774.40 
in  3  years  6  months  ? 

29.  In  what  time  will  $  6000  at  8  %  gain  an  interest  equal 
to  S  of  itself  ? 

30.  A  merchant  bought  a  bill  of  goods  amounting  to  $  1550, 
on  30  days'  credit,  but  could  have  bought  the  same  for  cash  at 
a  discount  of  5%.     What  was  the  difference  ? 

31.  What  is  the  difference  between  the  true  discount  and 
the  simple  interest,  both  at  5  %,  on  $  6415.50  for  3  years 
6  mouths  ? 


-/- 


REVIEW.  227 

32.  How  much  more  is  the  compound  interest  than  the 
simple  interest  of  $  1300  for  4  years  at  7  %  ? 

33.  On  a  note  for  $  2000,  dated  Jan.  1,  1880,  at  6  %  inter^ 
est,  there  was  paid,  July  1,  1880,  $  600.  Eequired  the  bal- 
ance due  Jan.  1,  1882. 

34.  A  man  makes  a  difference  in  his  income  of  $  82.50  by 
transferring  a  4  %  stock  at  92  to  a  5  %  stock  at  110.  What 
amount  was  transferred  ? 

35.  Invested  $  26250  in  bonds  at  87j^,  and  sold  the  same  at 
91.     What  was  the  gain  ? 

36.  Find  the  difference  between  the  income  derived  from 
$9080  invested  in  3%  stock  at  85 J,  and  that  derived  from 
$  9800  invested  in  5  %  bonds  at  122^. 

37.  A  owes  B  $  460,  of  which  $  100  is  to  be  paid  in  50  days, 
$  130  in  40  days,  and  the  remainder  in  140  days.  Find  the 
average  time. 

38.  A  merchant  bought  a  draft  on  St.  Louis  for  $  2660,  at 
60  days,  paying  $  2570.89.     What  was  the  rate  of  exchange  ? 

39.  A  note  for  $  500,  dated  Oct.  8,  1880,  and  bearing  inter- 
est at  6  %,  is  indorsed  as  follows  :  Nov.  4, 1881,  $  30  ;  Jan.  30, 
1882,  $  250.     What  will  be  due  July  1,  1882  ? 

40.  I  have  purchased  goods  to  the  amount  of  $  800,  on  a 
credit  of  4  months.  At  the  end  of  2  months  I  pay  $  100,  and 
at  the  end  of  3  months  I  pay  $  200.  How  long  after  the  ex- 
piration of  the  4  months  ought  the  balance  in  equity  to  remain 
unpaid  ? 

REVIEW  QUESTIONS. 

235.  What  is  percentage  1  254.  What  is  interest  ?  296.  What  is 
bank  discount  ?  277.  What  are  partial  payments  ?  284.  What  is 
compound  interest  ? 

270.  What  is  a  promissory  note  ?  '304.  What  are  bonds  1  What 
is  a  coupon  ?     306.  What  are  stocks  ?     309.  A  stock  certificate  ? 

316.  What  is  a  draft  ?  322.  What  is  exchange  ?  329.  How  are 
sterling  bills,  or  drafts,  quoted  1  332.  What  is  average,  or  equation 
of  payments  1    333.  What  is  the  average,  or  equated  time  ? 


228  RATIO   AND    PROPORTION. 

RATIO    AND    PROPORTION. 
RATIO. 

337.  1.  Thomas  has  20  books  and  John  5.  Thomas  has 
how  many  times  as  many  as  John  ? 

2.  Peter  is  16  years  old  and  his  brother  8.  How  do  their 
ages  compare  ? 

3.  What  part  of  27  feet  is  9  feet  ? 

4.  How  does  20  compare  with  5  ?    24  with  6  ?    27  with  9  ? 

5.  What  is  the  relation  of  $  35  and  $  7  ?  Of  51  miles  and 
17  miles  ?     Of  7  pounds  and  42  pounds  ? 

338.  Ratio  is  the  relation  of  two  like  numbers  shown 
by  their  quotient. 

It  is  determined  by  dividing  the  first  by  the  second. 

Thus, 

The  ratio  of  15  to  5  is  15  ~  5,  or  3. 

339.  Eatio  is  usually  indicated  by  :,  which  is  an  ab- 
breviated form  of  -^.     Thus, 

18  : 6  expresses  the  ratio  of  18  to  6. 

340.  The  Terms  of  a  ratio  are  the  two  numbers  com- 
pared. The  Antecedent  is  the  first  term  of  a  ratio,  the  Con- 
sequent  is  the  second  term,  and  the  two  terms  together  are 
called  a  Couplet. 

341.  An  Inverse  Ratio  is  a  ratio  formed  by  inverting  the 
terms  of  a  given  ratio.     Thus, 

8  :  9  is  the  inverse  of  9  :  8. 

342.  A  Simple  Ratio  is  the  ratio  of  two  numbers.  Thua> 

21  :  3  =  7  is  a  simple  ratio. 


RATIO   AND    PROPOETION.  229 

343.  A  Compound  Ratio  is  the  product  of  two  or  more 
simple  ratios.  It  is  usually  indicated  by  means  of  the 
brace.     Thus, 

8  :  2) 

9:3>:  or  8x9X10:  2x3X5  is  a  compound  ratio 
10  :  5) 
aqual  to  the  simple  ratio  720  :  30. 

A  compound  ratio  may  be  changed  to  a  simple  ratio  by 
muUvplying  antecedents  together  for  a  new  antecedent,  and 
consequents  for  a  new  consequent. 

A  ratio,  like  a  fraction,  is  simply  an  indicated  division 
(Art.  108).  The  principles  of  common  fractions  are  equally 
applicable  to  ratios,  the  antecedent  being  the  numerator 
and  the  consequent  the  denominator  (Arts.  Ill,  114). 

344.    Principles  of  Ratio. 

1.  The  ratio  is  equal  to  the  antecedent  divided  by  the  con- 
sequent. 

2.  The  consequent  is  equal  to  the  antecedent  divided  by  the 
ratio. 

3.  The  antecedent  is  equal  to  the  consequent  multiplied  by 
the  ratio. 

4.  Multiplying  or  dividing  both  the  antecedent  and  the  con- 
sequent by  the  same  number  does  not  change  the  ratio. 


EXERCISES. 

ind 

1  the  ratios  of 

6. 

65  :  15. 

9.   5:i. 

12. 

#:f 

7. 

25  :  625. 

10.  2.25  :  0.75. 

13. 

63  :  72. 

a 

$  256  :  $  228. 

11.   3i:13. 

14. 

6J  :  7|. 

15.  What  is  the  inverse  ratio  of  Q^  :  15  ? 

16.  What  is  the  inverse  ratio  of  25  :  625  ? 

17.  Which  is  the  greater  ratio,  12  to  13  or  25  to  27  ? 


230  RATIO   AND    PROPORTION. 

18.  If  6.25  is  the  antecedent  and  5  the  ratio,  what  is  th^ 
consequent  ? 

19.  If  27  is  the  consequent  and  ^  the  ratio,  what  is  the 
antecedent  ? 

20.  What  is  the  value  of  the  compound  ratio  o  !  ^  [■  ? 

21.  What  is  the  ratio  compounded  of  (  6  ;  8)  X  (16  :  10) 
X  (12  :  9)  ? 

PROPORTION. 

22.  The  ratio  of  21  to  7  is  what  number  ?     Of  51  to  17  ? 

23.  Name  two  numbers  having  the  same  ratio  as  51  to  17. 

24.  What  number  has  the  same  relation  to  17  as  21  has 

tK)7? 

25.  If  12  yards  of  cloth  cost  $  40,  what  part  of  $  40  will 
3  yards  cost  ? 

26.  How  does  the  ratio  of  3  yards  to  12  yards  compare  with 
the  ratio  of  $  10  to  $  40  ? 

345.  A  Proportion  is  an  equality  of  ratios.     Thus, 

12:3  =  40:10isa  proportion. 

The  equality  of  ratios  may  be  indicated  either  by  = 
or  :  :.     Thus, 

8  :  2  =  16  :  4,"  or  8  :  2  :  :  16  :  4. 

means  8  to  2  equals  16  to  4,  or  8  is  to  2  as  16  is  to  4. 

346.  Each  term  of  a  proportion  is  called  a  Proportional ; 
the  first  and  fourth  terms  are  called  Extremes;  and  the 
second  and  third  terms,  Means. 

When  the  two  means  are  the  same  number,  that  number 
is  a  Mean  Proportional  between  the  two  extremes.     Thus, 

In  12:6  =  6:  3,  6  isa  mean  proportional  between  12 
and  3. 


RATIO 

AND 

PROPORTION. 

347. 

In  the 

proportion 

6 

:  3  = 

=  4: 

2, 

as 

;  the  ratios  are 

1  equal,  we  have 

6 
3  ' 

4 

~  2' 

231 


Changing  these  fractions  to  a  common  denominator,  we 
have 

6x23X4 


As  these  fractions  are  equal  and  their  denominators 
alike,  their  numerators  must  be  equal,  or  6  X  2  =  3  X  4. 
But  6  and  2  are  the  extremes,  and  3  and  4  the  means. 
Hence  the  following 

348.    Principles  of  Proportion. 

1.  I?i  a  proportion  the  product  of  the  means  is  equal  to  the 
product  of  the  extremes, 

2.  Either  extreme  is  equal  to  the  product  of  the  means  di- 
vided hy  the  other  extreme. 

3.  Either  mean  is  equal  to  the  product  of  the  extremes  di- 
vided hy  the  other  mean. 

WRITTEN    EXERCISES. 

Find  the  missing  term  represented  by  x  in  the  following 
proportions  : 

27.  14  :  7  =:  18  :  iK.  31.  |^  :  cc  :  :  4  :  8. 

28.  5  :  20  =  cc  :  60.  32.  $45  :  $24  :  :  15  yd.  :  x. 

29.  cc  :  8  =  65  :  13.  33.  ir  :  $  9  :  :  60  men  :  18  men. 

30.  648  :  243  ==  24  :  x.  34.  5  tons  :  J  ton  :  :  x  :  $7.50. 


232  KATIO   AND    PROPORTION. 


SIMPLE    PROPORTIOlSr. 


349.  A  Simple  Proportion  is  an  equality  between  two 
simple  ratios. 

It  applies  to  the  solution  of  questions  in  which  three 
terms  of  a  proportion  are  given  to  find  the  fourth. 

Note.  —  Of  the  given  terms  two  must  be  of  the  same  kind,  and  constitute  a 
ratio  ;  and  the  other  must  be  of  the  same  kind  as  the  required  term,  and  con- 
stitute with  it  another  ratio  equal  to  the  first. 

WRITTEN    EXERCISES. 

35.  If  37  yards  of  cloth  cost  $  111,  what  will  19  yards  cost  ? 

37  :  19  =  $  111  :  $  x  Solution.  —  As  37  yards  must  evi- 

o  dently  have  the   same   ratio   to   19 

19  X  $111  yards  that  111  1,  the  cost  of  37  yards, 

— =  $  57         has  to  the  cost  of  19  yards,  or  the 

answer,  we  arrange  the  terms  so  as 
to  express  the  equality  of  these  ratios.     Or, 

As  the  fourth  term  is  to  be  dollars,  we  make  $  111  the  third  term. 
The  fourth  term  is  to  be  less  than  the  third  term,  because  19  yards 
will  cost  less  than  37  yards.  Hence  the  second  term  must  be  smaller 
than  the  first,  and  the  first  ratio  is  37  :  19.  Dividing  the  product  of 
the  means  by  the  given  extreme,  we  have  as  the  answer  1 57.     Or, 

If  37  yards  cost  I  111,  1  yard  costs  ^  of  $  111,  and  19  yards  19 
times  as  much,  or  J-|  of  $  111,  or  $  57. 

36.  If  12  barrels  of  apples  cost  $  51,  what  will  30  barrels 
cost? 

37.  If  the  rent  of  183  acres  of  land  is  $  273,  what  will  be 
the  rent  of  61  acres  ? 

38.  What  number  of  men  will  be  required  to  perform  in  16 
days  a  piece  of  work  that  would  take  30  men  48  days  ? 

39.  If  24  men  can  mow  a  field  in  15  days,  how  many  days 
will  it  take  20  men  to  dp  it  ? 


RATIO   AND    PROPORTION.  233 

350.    Rule  for  Simple  Proportion. 

Make  that  number  which  is  of  the  same  kind  as  the  answer 
the  third  term.. 

If  from  the  nature  of  the  question  the  answer  is  to  he 
larger  than  the  third  term^  make  the  larger  of  the  remaining 
numbers  the  second  and  the  smaller  the  first  term  ;  but  if  the 
answer  is  to  be  smaller  than  the  third  term,  make  the  second 
term  smaller  than  the  first. 

Divide  the  product  of  the  means  by  the  given  extreme,  and 
the  quotient  is  the  foiorth  term,  or  answer. 

40.  When  $  120  are  paid  for  15  barrels  of  flour,  what  will 
79  barrels  cost  ? 

41.  If  7  gallons  of  molasses  cost  $  5.88,  what  will  27  gal- 
lons cost  ? 

42.  If  a  man  travel  319  miles  in  11  days,  how  far  will  he 
travel  in  47  days  ? 

43.  If  27  men  can  do  a  piece  of  work  in  12  days,  how  long 
will  it  take  36  men  to  do  it  ? 

44.  Find  the  cost  of  7  sheep  when  98  cost  $  441. 

45.  What  time  should  24  men  take  to  perform  a  piece  of 
work  which  18  men  can  perform  in  15  days  ? 

46.  A  garrison  of  2100  men  has  provisions  for  9  months, 
but  receives  a  reinforcement  of  600  men.  How  long  will  the 
provisions  last  ? 

47.  If  4|  bushels  of  oats  cost  $2^,  what  will  19J  bushels 
cost  ? 

48.  74  men  had  provisions  for  35  days,  but  after  five  days 
20  men  were  sent  away.  How  long  will  the  provisions  last 
the  remaining  54  men  ? 

49.  If  6336  stones  of  3i  feet  in  length  will  make  a  certain 
quantity  of  wall,  how  many  similar  stores  of  2f  feet  in  length 
will  make  a  like  quantity  ? 


234  RATIO   AND   PROPORTION. 

50.  If  3f  tons  of  coal  cost  $  27.50,  what  will  4f  tons  cost  ? 

51.  A  certain  piece  of  work  was  to  have  been  performed  by 
288  men  in  72  dajs,  but,  a  number  of  them  having  been  sent 
away,  it  was  performed  in  108  days.  What  was  the  number  of 
men  sent  away  ? 

52.  If  3^  cords  of  wood  cost  $  11.37|,  what  will  12|  cords 
cost  ? 


COMPOUND    PROPORTION. 

351.    A  Compound  Proportion  is  an  equality  between  a 
compound  and  a  simple  ratio.     Thus, 

3:4; 


^  ^  J       45  :  96  is  a  compound  proportion. 

It  applies  to  the  solution  of  questions  which  would  re- 
quire several  simple  proportions. 

WRITTEN    EXERCISES. 

53.   If  4  men  can  earn  $  64  in  8  days,  how  much  can  12 
men  earn  in  3  days  ? 

4  :  12 )  Solution.  —  As  the  answer  sought 

8  :     3|  =  ^^"^  •  ^^      is  in  dollars,  we  make  the  $64  the 

Qj.  third  term,  and  cc,  representing  the 

o  o  answer,  the  fourth  term.     If  the  an- 

T9  V  ^  V  (M  ^^^^  depended  only  on  the  number 

^ A^  ~  ^^         ^^  "^^"'  ^*  would  be  larger  than  the 

^  '^  ^  third  term,  as  12  men  will  earn  more 

than  4  men  ;  hence  the  first  ratio  is  4  to  12.  But  if  the  answer  de- 
pended only  on  the  number  of  days  worked  it  would  be  smaller  than 
the  third  term,  as  less  can  be  earned  in  3  days  than  in  8  days  ;  hence 
the  second  ratio  is  8  to  3. 

Dividing  the  product  of  the  means  by  the  product  of  the  given  ex- 
tremes,  we  have  $  72  as  the  answer. 

Or,  if  4  men  can  earn  $64  in  8  days,  12  men  in  the  same  time 
can  earn  y  of  5^  64,  and  in  3  days  |  of  l^  of  $  64,  or  $  72. 


RATIO   AND    PROPORTION.  235 

352.    Rule  for  Compound  Proportion. 

Make  that  number  which  is  like  the  answer  the  third  term. 
Form  a  ratio  of  each  joair  of  the  remaining  numbers  of  the 
same  kind  according  to  the  rule  for  simple  proportion^  as  if 
the  answer  depended  on  them  alone.  Divide  the  product  of 
the  means  by  the  product  of  the  given  extremes,  and  the  quo- 
tient is  the  fourth  term,  or  answer. 

54.  If  3  men  can  make  108  pairs  of  shoes  in  2  days,  how- 
many  pairs  can  2  men  make  in  a  week  ? 

55.  If  $  250  yields  $  175  interest  in  7  years,  how  long  will 
it  take  $  500  to  yield  $  360  at  the  same  rate  ? 

56.  If  24  men  can  reap  76  acres  in  6  days,  how  many  men 
will  reap  114  acres  in  9  days  ? 

57.  How  many  acres  can  10  men  plow  in  14  hours,  if  5 
men  plow  6  acres  in  lOJ  hours  ? 

58.  Two  cogged  wheels,  one  of  which  has  15  cogs  and  tht 
other  28,  work  in  each  other.  If  the  first  turns  16  times  in  7^ 
seconds,  how  often  will  the  other  turn  in  4  seconds  ? 

59.  If  15  men  are  fed  for  7  days  when  flour  is  $  8  a  barrel, 
what  must  be  the  price  when  '10  men  are  fed  8  days  at  the 
same  cost  ? 

60.  If  a  man  travels  117  miles  in  15  days,  employing  only 
9  hours,  how  far  would  he  go  in  20  days,  traveling  12  hours 
a  day  ? 

61.  If  96  horses  eat  192  tons  of  hay  in  one  winter,  how 
many  tons  will  150  horses  eat  in  6  winters  ? 

62.  If  a  man,  walking  12  hours  each  day,  travels  250  miles 
in  9  days,  in  how  many  days,  walking  10  hours  each,  at  the 
same  rate,  would  he  travel  400  miles  ? 

63.  If  the  expenses  of  a  family  of  8  persons  amount  to 
$  84  in  16  weeks,  how  long  will  $  200  support  a  family  of  6 
persons  ? 

j4.  If  a  pasture  of  16  acres  will  feed  6  horses  for  4  months, 
how  many  acres  will  feed  12  horses  for  9  months  ? 


236  RATIO   AND    PROPORTION. 

65.  If  1080  bricks,  8  inches  long  and  4  inches  wide,  are  re- 
quired for  a  walk  20  feet  long  and  6  feet  wide,  how  many 
bricks  will  be  required  for  a  walk  100  feet  long  and  4  feet 
wide  ? 

66.  If  34  men  can  saw  90  cords  of  wood  in  6  days,  when 
the  days  are  9  hours  long,  how  many  cords  can  8  men  saw  in 
36  days  when  they  are  12  hours  long  ? 

67.  If  12  men  in  15  days  can  build  a  wall  30  feet  long, 
6  feet  high,  and  3  feet  thick,  working  12  hours  a  day,  in  what 
time  will  30  men  build  a  wall  300  feet  long,  8  feet  high,  and 
6  feet  thick,  working  8  hours  a  day  ? 

68.  If  a  loaf  which  sells  for  20  cents  when  wheat  is  $  4  a 
bushel,  weighs  3  pounds,  what  is  the  price  of  wheat  when  a 
12-cent  loaf  weighs  2J  pounds  ? 

69.  If  a  bin  8  ft.  long,  41  ft.  wide,  and  2|  ft.  deep,  holds  67| 
bu.,  how  deep  must  another  bin  be  made,  that  is  18  ft.  long , 
and  3f  ft.  wide,  to  hold  450  bu.  ? 

70.  If  the  annual  salary  of  a  man  who  works  8  hours  a  day, 
48  weeks  in  the  year,  is  $  1200,  how  much  ought  a  conductor 
to  receive  per  month  who  works  14  hours  daily  the  year 
round  ? 

QUESTIONS. 

338.  What  is  ratio  1  338.  How  is  it  determined  ?  340.  What  are 
the  terms  of  a  ratio  1 

341.  What  is  an  inverse  ratio?  342.  A  simple  ratio?  343.  A 
compound  ratio  ?     344.  What  are  the  principles  of  ratio  ? 

345.  What  is  a  proportion  ?  346.  What  is  each  term  of  a  propor- 
tion called  ?  What  is  a  mean  proportional  ?  348.  What  are  the 
principles  of  proportion  ? 

349.  What  is  a  simple  proportion  1  350.  Which  number  is  made 
the  third  term  1  How  are  the  terms  arranged  1  How,  then,  is  the 
required  extreme  found  ? 

351.  What  is  a  compound  proportion  ?  352.  Which  term  is  made 
the  third  term  ?  How  is  each  pair  of  the  remaining  numhers  ar- 
ranged ?    How,  then,  is  the  required  term  found  ? 


PARTNERSHIP.  237 


PARTNERSHIP. 

353.  1.  Two  men  share  between  them  $35,  the  one  re- 
ceiving $  3  as  often  as  the  other  $  4.  How  much  does  each 
receive  ? 

2.  What  number  is  f  of  35  ?     f  of  35  ? 

3.  A  and  B  are  in  Business  together.  A  put  in  $  3000  and 
B  $  5000.  They  gain  $  800.  What  are  their  respective  shares 
of  it? 

4.  Divide  $  800  into  two  parts  having  the  ratio  of  3  to  5. 

5.  Divide  54  oranges  between  two  boys  in  the  ratio  of  4 
to  5. 

354.  Partnership  is  the  association  of  two  or  more  per- 
sons in  business. 

355.  The  Company,  or  Firm,  is  the  association,  and  the 
Partners  are  the  members. 

356.  The  Capital,  or  Stock,  is  that  which  is  invested  in 
the  business,  and  the  Dividend  is  the  profits  shared  by  the 
partners. 

The  profits  or  losses  are  usually  shared  according  to  the 
terms  of  the  agreement,  or  contract,  made  when  the  part- 
nership is  formed. 

In  the  absence  of  a  special  agreement,  dividends  are  in 
proportion  to  the  capital  invested  and  the  time  during 
which  it  is  invested. 

WRITTEN    EXERCISES. 

6.  A,  B,  and  C  engage  in  trade.  A  furnishes  $  200,  B  $  250, 
and  C  $350.  They  gain  1100.80.  What  is  each  partner's 
share  of  the  gain  ? 


238  PARTNERSHIP. 


Solution, 


$200  +  $250  +  $350  :r=  $800,  the  capital. 
A's  stock  =  f  g-g  =  1 ;  1  of  $  100.80  =:  $  25.20,  A's  gain. 
B's     "      =  li%  =  j^ ;  j%  of  $  100.80  =  $  31.50,  B's  gain. 
C's     "      =  |5^  =  tV  ;   i\  of  $  100.80  =  $  44.10,  C's  gain. 

As  A's  stock  is  ^  of  the  entire  capital,  B's  stock  y^g,  and  C's  ^^^ 
A  must  have  J,  B  ^^g,  and  C  ^^  of  the  gain.  Hence  A's  gain  is 
$  25.20,  B's  $  31.50,  and  C's  $  44.10. 

7.  A,  B,  and  C  engage  in  trade.  A  puts  in  $  6000,  B  $  9000, 
and  C  $5000.  They  gain  $1680.  What  is  each  partner's 
share  of  the  gain  ? 

357.    To  find  the  gain  or  loss  of  partners, 

Rule. 

Take  for  each  partner  such  a  part  of  the  gain  or  loss  as  his 
stock  -is  of  the  entire  capital. 

Note.  —  The  rule  applies  to  the  distribution  of  the  assets  of  bankrupts  and 
other  like  apportionments- 

8.  A,  B,  and  C  engage  in  trade,  investing  capital  to  the 
amount  of  $1280,  $1760,  and  $1920,  respectively.  Their 
profits  were  $  2790.     How  were  they  divided  ? 

9.  A  bankrupt  owes  three  creditors.  A,  B,  and  C,  $  1750, 
$  2100,  and  $  2650,  respectively.  His  assets  are  $4225.  What 
should  they  each  receive  ? 

10.  Hall  and  Bishop  gain  by  trade  $  728.  Hall  put  in 
$  1200,  and  Bishop  $  1600.     What  is  the  gain  of  each  ? 

13.  B  puts  on  board  of  a  ship  400  barrels  of  flour,  C  600, 
and  D  400 ;  but  when  at  sea  it  was  found  necessary  to  throw 
360  barrels  overboard.  How  much  of  the  loss  should  fall  to 
each  ? 

12.  A,  B,  and  C  hire  a  pasture  for  $  300.  A  puts  in  8 
horses,  B  6,  and  C  10.     How  many  dollars  should  each  pay  ? 


PAKTNERSHIP.  239 

13.  A,  B,  and  C  engage  in  trade.  A  puts  in  $  1400,  B  $  600, 
and  C  125  barrels  of  Hour.  They  gained  $  180 ;  of  which  C 
took  $60  as  his  part.  What  will  A  and  B  receive,  and  what 
was  the  value  of  C's  flour  a  barrel  ? 

358.  When  the  capital  of  the  partners  is  employed  for 
unequal  times, 

Find  the  product  of  each  partner's  stock  multiplied  hj  the 
time  it  was  invested,  and  divide  the  gain  or  loss  in  propor- 
tion to  the  products. 

14.  A,  B,  and  C  engage   in  trade.     A  puts  in   $300  for 

7  months,  B  $  500  for  8  months,  and  C  $  200  for  12  months. 
They  gain  $  170.     What  is  each  man's  share  of  the  gain  ? 

Solution, 

A's  $  300  for    7  months  =  $  2100  for  1  month. 

B's  $  500  for    8  months  =     4000  for  1  month. 

C's  $200  for  12  months  =     2400  for  1  month. 

The  entire  stock  is  the  same  as  $  8500  for  1  month. 

m%  =  ti ;  Si  of  $  170  =  $  42,  A's  gain. 
m%  =  tS ;  t»  of  *  170  =  $  80,  B's  gain. 
im  -  SI  5  If  of  $  170  -  $  48,  C's  gain. 

As  $300  for  7  months  is  the  same  as  $  2100  for  1  month,  $500  for 

8  months  the  same  as  $4000  for  1  month,  and  $200  for  12  months 
the  same  as  $  2400  for  1  month,  the  entire  stock  is  $  2100  +  $  4000  + 
%  2400  =  $  8500  for  1  month. 

Hence  A's  gain  will  be  |^  of  the  whole  gain,  or  $  42  ;  B's,  |^,  or 
$80;  andC's,  II,  or  $48. 

15.  A,  B,  and  C  had  a  joint  stock  of  $  2400.  A's  part  was 
$  750,  and  continued  in  trade  4  months ;  B's  was  $  850,  and 
continued  8  months  ;  the  remainder  was  C's,  and  continued  in 
trade  throughout  the  year.  They  lost  $  640.  What  was  each 
man's  share  of  it  ? 


240  PARTNERSHIP. 

16.  A,  B,  and  C  were  in  partnership.  A  liad  in  the  busi- 
ness $  5000  for  8  months,  B  $  4000  for  12  months,  and  C 
$  3000  for  15  months.  The  profits  were  $  1330.  How  much 
is  each  partner's  part  of  the  profits  ? 

17.  A  and  B  rent  a  pasture  for  $46.80.  A  puts  in  30 
horses  for  33  days,  and  B  21  horses  for  42  days.  How  much 
ought  each  to  pay  of  the  rent  ? 

18.  A,  B,  and  C  form  a  partnership.  A  furnishes  $  500  foi 
9  months,  B  $  700  for  1  year,  and  C  $  400  for  15  months. 
They  lose  $  300.     What  is  each  man's  share  of  the  loss  ? 

19.  A  and  B  are  in  partnership.  A  put  in  $  6000,  and  at 
the  end  of  6  months  put  in  $4000  more;  B  put  in  $12000, 
and  at  the  end  of  8  months  took  out  $  6000.  They  trade  1 
year,  and  gain  $2160.  What  is  each  man's  share  of  the 
gain  ? 

20.  A  and  B  enter  into  partnership  for  1  year.  A  had 
$  500  in  the  business  during  the  first  4  months,  and  $  300 
more  during  the  remainder  of  the  year ;  whereas  B  had  only 
$400  during  the  first  6  months,  but  $900  during  the  last 
6  months.  They  gained  $  2400.  What  was  each  man's  share 
of  the  gain  ? 

21.  Jan.  1,  1881,  Wood  goes  into  business  with  a  capital 
of  $  6000.  March  1,  Furbush  joins  him  witli  $  5000.  eJuly  1, 
they  take  Davis  into  the  partnership  with  $4000  capital, 
agreeing  to  pay  him  6  %  interest  for  his  money,  and  give  him 
an  annual  salary  of  $  1200.  The  profits  were  $  3160,  out  of 
which  Davis  was  paid  and  the  balance  divided  between  the 
other  partners.     Find  each  man's  share  at  the  end  of  the  year. 

22.  Jan.  1,  1882,  A  and  B  form  a  partnership  for  a  year. 
A  furnishes  $2000,  and  B  $3000.  May  1,  they  take  C  into 
the  firm  with  a  capital  of  $  5000.  August  1,  A  receives  a  leg- 
acy of  $4000,  which  he  adds  to  the  capital.  Oct.  1,  B  with- 
draws $  1000  of  his  capital.  At  the  end  of  the  year  the  firm's 
net  gains  were  $  5850.     Divide  it  equitably  among  the  part- 


INVOLUTION   AND    EVOLUTION.  241 

INVOLUTION   AND    EVOLUTION. 
INVOLUTION. 

359.  1.   What  is  the  product  of  5  used  twice  as  a  factor  ? 

2.  Of  what  number  are  5  and  5  the  factors  ? 

3.  What  is  the  product  of  5  used  three  times  as  a  factor  ? 

4.  Of  what  number  are  5,  5,  and  5  the  factors  ? 

5.  What  is  the  product  of  .3  used  three  times  as  a  factor  ? 
Of  I  used  three  times  as  a  factor  ? 

360.  A  Power  of  a  number  is  the  product  arising  from 
taking  the  number  a  certain  number  of  times  as  a  factor. 

The  First  Power  of  a  number  is  the  number  itself ; 
The  Second  Power  of  a  number  is.  the  product  arising 
from  taking  the  number  twice  as  a  factor ; 

The  Third  Power  of  a  number  is  the  product  arising 
from  taking  the  number  three  times  as  a  factor ;  and  so 
on.     Thus, 

The  first  power  of  3  is  3. 
"     second     "       3  is  3  x  3,  or  9. 
"     third        "       3  is  3  X  3  X  3,  or  27. 
"     fourth      "       3  is  3  X  3  X  3  X  3,  or  81. 

The  second  power  is  also  called  the  Square  of  the  num- 
ber, as  the  area  of  a  square  is  the  product  of  two  equal  fac- 
tors. 

The  third  power  is  called  the  Cuhe,  as  the  volume  of  a 
cube  is  the  product  of  three  equal  factors. 

361.  The  Exponent  of  a  power  is  a  small  figure  placed 
at  the  right  and  above  a  number.     Thus, 

25^  means  the  second  power  or  square,  of  25 ; 
3.1^  means  the  third  power,  or  cube,  of  3.1 ; 
(I)*  means  the  fourth  power  of  |. 


242  INVOLUTION   AND    EVOLUTION. 

362.    Involution  is  the  process  of  finding  powers. 

WRITTEN    EXERCISES. 

6.  What  is  the  third  power  of  12  ? 

Solution.  —  12^  =z  12  X  12  X  12  =  1728. 

7.  Find  the  squares  and  cubes  of  the  first  nine  numbers. 

363.     Rule  for  Involution. 

Use  the  given  number  as  many  times  as  a  factor  as  thero 
are  units  in  the  exponent  of  the  required  power. 

rind  the  powers  indicated  by  the  exponents : 

8.  231  11.    (1)2.  14.  25^  17.  3.52. 

9.  163.             12.  .(2J)3.          15.    Ill             18.  (141)2 
10.    13^              13.   3.6".             16.    0.151           19.  0.073. 

EVOLUTION. 

20.  What  are  the  factors  of  9  ?     Of  25  ?     Of  49  ? 

21.  What  are  the  two  equal  factors  of  16  ?    Of  64  ?    Of  81  ? 

22.  What  are  the  three  equal  factors  of  27  ?     Of  64  ? 

364.  A  Root  of  a  number  is  one  oi  the  equal  factors 
which  produce  it. 

The  Second,  or  Square,  Root  of  a  number  is  one  of  the 
two  equal  factors  which  produce  it ; 

The  Third,  or  Cidje,  Root  of  a  number  is  one  of  the  three 
equal  factors  which  produce  it ; 

The  Fourth  Root  of  a  number  is  one  of  the  four  equal 
factors  which  produce  it ;  and  so  on. 

365.  A  Perfect  Power  is  a  number  whose  exact  root  can 
be  found,  and  an  Imperfect  Power  is  a  number  whose  root 
cannot  be  exactly  tound. 


INVOLUTION   AND    EVOLUTION.  243 

366.    The  Radical  Sign,  y/,  is  used  to  indicate  a  root. 


Thus. 


V  16  means  the  second,  or  square,  root  of  16. 


Si 


V  25  means  the  third,  or  cube,  root  of  25. 


The  number  in  the  opening  of  the  sign,  called  the  Index 
of  the  root,  denotes  the  name  of  the  root. 

The  index  of  the  square  root  may  be  understood.  Thus, 
V  81,  or  V  81,  may  indicate  the  second,  or  square,  root  of  81. 

367.  Evolution  is  the  process  of  finding  roots.  It  is 
the  reverse  of  involution. 

SQUARE  ROOT. 

368.  To  extract  the  square  root  of  a  number  is  to  find 
one  of  the  two  equal  factors  which  produce  it. 

369.  The  square  of  a  number  contains  twice  as  many 
figures  as  the  root,  or  twice  as  mo.ny  less  one.     Thus, 

12  =1.  92  ==  81. 

102  ^  i/QO.  992  =  98^01. 

1002  ^  i/oO'OO.  9992  ==  99'80'01. 

370.  If  a  povMr  is  separated  into  periods  of  two  figures 
each,  beginning  at  the  decinfial  point,  the  numher  of  periods 
will  show  the  number  of  figures  in  the  root.     Thus, 

The  square  root  of  2'35.'92'96  contains  two  integral  and 
two  decimal  figures. 

371.  The  square  of  the  highest  order  of  units  in  the  root 
is  found  in  the  highest  period  of  the  power.     Thus, 

9  tens2,  or  902  =  81  hundreds,  or  81'00. 

9  hundreds2,  or  9002     —  81  ten-thousands,  or  81'00'OQ 

9  thousands^,  or  90002=  81  millions,  or  81'00^00'00. 


244  INVOLUTION   AND   EVOLUTION. 

372.  The  parts  which  make  up  a  second  power  may  be 
learned  by  a  careful  inspection  of  the  process  of  multipli- 
cation by  which  the  power  is  produced.  For  example,  let 
us  square  36,  keeping  its  tens  and  ones  and  their  products 
distinct  and  separate.     Thus, 

36  =  30  +  6  =  tens  +  onea 

36  =  30  4-  6  =  tens  +  ones. 

2l6=  (30x6) +  62=  (tens  X  ones)  +  ones^ 

108    =     30^  +  (30  X  6)  =  tens^  +  (tens  X  ones). 

1296  =  802  -f-  2  (30  ><  g)  +  59  ^  tens^  +  2  tens  X  ones  +  ones^. 

That  is,  the  square  of  any  number  composed  of  tens  and 
ones  equals 

373.  The  square  of  tlie  tens,  plus  two  times  the  tens  times 
the  ones,  plus  the  square  of  tlu  ones,  or  t^  +  2t  X  0  +  0^. 

WRITTEN     EXERCISES. 

23.   Find  the  square  root  of  1296. 

12^96  (30  +  6  Solution.  —  This 

302  — -    goo  :=  t2  power  contains  two 

2t=:2x30:=:60"396  =  2tXo  +  o2  Periods  ;   hence  its 

oor\ 9  f  V  o  ^°°*  ^^^  ^^^  figures, 

tens  and  ones  (Art. 
370).  The  square 
of  the  tens  is  in  the 


36  =  o2 
o2  =  6  X  6  ==      36 


highest  period  (Art.  371).  Taking  out  of  the  12  hundreds  the 
largest  "tens 2"  possible,  900,  and  placing  its  root,  3  tens,  or  30,  at 
the  right,  there  remains  396,  which  must  be  the  "  2  tens  X  ones  + 
ones*"  (Art.  372).  The  "ones*"  being  but  a  small  part  of  the  396, 
we  may  treat  this  number  as  the  approximate  product  of  the  "  2  tens 
X  ones."  Dividing  this  product,  396,  by  one  of  its  factors,  "  2  tens," 
or  60,  we  have  6  as  the  other  factor,  the  "ones."  Taking  from  396 
the  "  2  tens  X  ones,"  or  60  X  6,  or  360,  there  remains  36,  which  con^ 
tains  the  "  ones*."  Taking  the  ones*,  or  6*,  from  36,  nothing  remain^ 
Hence  we  conclude  that  30  +  6,  or  36,  is  the  root  required. 


INVOLUTION   AND   EVOLUTION. 


245 


Geometrical  Explanation  of  Square  Root. 

374.  As  the  length  of  a  square  is  the  square  root  of  its 
ftrea,  the  method  of  extracting  the  square  root  of  a  number 
may  be  illustrated  by  the  process  of  finding  the  length  of  a 
square,  its  area  being  given. 


It  is  required  to  find  the  length  of  a  square  containing 
1296  square  inches. 


Solution, 


A  square  containing  1296  sq.  in.  cannot  be  40  in.  long,  for  a  40-inch 
square  contains  1600  sq.  in.  It  must  be  more  than  30  in.  long,  for  a 
30-inch  square  contains  but  900  sq.  in.  The  length  of  the  given  square 
must  therefore  be  between  30  and  40  inches. 


Removing  from  the  given  square.  A,  a  30-inch  square,  B,  contain- 
ing 900  sq.  in.,  there  remains  a  surface  containing  396  sq.  in.,  largely 
made  up  of  the  rectangles  G  and  Z),  whose 
length  is  evidently  that  of  the  square,  B.  It 
is  obvious  that  the  width  of  these  rectangles 
added  to  the  length  of  the  square,  B,  will  give 
the  required  length  of  the  given  square,  A. 
Now  the  width  of  a  rectangle  is  found  by  di- 
viding its  area  by  its  length  (Art.  218).  The 
length  of  each  of  these  rectangles,  G  and  D,  is 
30  in.,  and  their  united  length  2  X  30  in.,  or  60  inches.  Dividing 
their  approximate  area,  396  sq.  in.,  by  their  length,  60  in.,  we  have 
as  their  probable  width  6  inches. 


246 


INVOLUTION   AND    EVOLUTION. 


Removing  the  rectangles  C  and  D,  there 
remains  the  little  square,  E,  whose  length  is 
evidently  the  width  of  the  rectangles  removed. 
Combining  the  two  rectangles  and  the  little 
square,  we  find  their  united  length  to  be  60  -{- 
6,  or  66  in.  Multiplying  their  length  and 
width  together,  we  find  their  area  to  be  Q^  X  6, 
or  396  sq.  in.,  the  exact  area  of  that  portion 
of  the  given  square.  A,  remaining  after  the 
removal  of  the  square,  B. 

We  therefore  conclude  that 
the  length  of  the  given  square 
is  36  inches. 


The  work  may  be  expressed  thus : 


Length. 

30  in.  X  2  =  60  in 
6  " 


Area.  Width. 

12'96  sq.  in.  (30  in.  +  6  in. : 
.  900     " 


:  36  in. 


66 


396 


396 


The  process  may  be  shortened  by  the  omission  of  the  ciphers,  and 
proved  by  involution. 

24.   Extract  the  square  root  of  3998.64. 


39'98.'64  (63.234+ 
36 


123 

1262 

12643 


398 
369 


2964 
2524 


126464 


44000 

37929 _ 
607100 
505856 
101244 


INVOLUTION   AND   EVOLUTION.  247 

375.     Rule  for  finding  the  Square  Root  of  a  Number. 

Beginning  at  the  decimal  point,  sejparate  the  given  number 
into  periods  of  two  figures  each. 

Find  the  greatest  square  in  the  left  period,  and  place  its 
root  at  the  right ;  subtract  the  square  of  this  root  from  the 
first  period,  and  to  the  remainder  annex  the  next  period  for  a 
dividend. 

Divide  this  dividend,  omitting  the  last  figure,  by  double  the 
root  already  found,  and  annex  the  quotient  to '  the  root  and 
also  to  the  divisor. 

Multiply  the  divisor  as  it  now  stands  by  the  last  root  figure, 
and  subtract  the  product  from  the  diaidend. 

If  there  are  more  pjeriods  to  be  brought  down,  proceed  in  the 
same  manner  as  before. 

Note  1.  —  If  0  occurs  in  the  root,  annex  0  to  the  divisor  and  another  period 
to  the  dividend,  and  proceed  as  before. 

Note  2.  —  If  there  is  "a  remainder  after  using  all  the  periods,  w^e  can  only  ap- 
proximate to  the  root.  But  nearer  and  nearer  approximations  can  be  obtained 
by  annexing  and  using  successive  periods  of  decimal  ciphers. 

25.   What  is  the  square  root  of  49.434961  ?      ' 
49.'43'49'61  (7.031 

^" Here  in  the  process,  as  0  occurs  in 

4349  the  root,  we  annex  0  to  the  divisor, 

4209  14,  and  annex  the  next  period  to  the 


1403 
14061 


14061  corresponding  dividend. 

14061 


Find  the  square  root 

26.  Of  9216.  31.  Of  6.7081.  36.   Of  0.9409. 

27.  Of  27225.    32.  Of  4.2025.  37.  Of  77841. 

28.  Of  182329.    33.  Of  1866.24.  38.  Of  16.2409. 

29.  Of  717409.    34.  Of  0.009409.  39.  Of  14.8996. 

30.  Of  948676.    35.  Of  0.05625.  40.  Of  39.0625. 


248  INVOLUTION    AND    EVOLUTION. 

41.  What  is  the  square  root  of  538  to  the  nearest  hun* 
dredth  ? 

42.  What  is  the  value  of  V^  to  the  nearest  thousandth  ? 


43.  What  is  the  value  of  ^0.002  to  the  nearest  ten-thou- 
sandth ? 

376.    To  extract  the  root  of  a  fraction, 

First  change  it  to  smallest  terms ;  then,  if  both  terms  are 
perfect  squares,  take  the  root  of  each ;  otherwise  change  the 
fraction  to  a  decimal,  and  find  the  root. 

Mixed  numbers  may  he  clianged  either  to  improper  frac- 
tions or  to  mioced  decimals. 

44.  What  is  the  square  root  of  5*^^  ? 

Solution.  —  t/-__  —  ' ^:3  — 

V529      v/629      23 

Find  the  value  of  the  following : 

*^-  ^HU-  *^-  s/49f.  53.  v/|  +  J  +  |. 

*6-  ^lim-  50.  y/^.  54.  v/98l|. 

47.  y/go^.  51.  y'72i.  55'  vTJef. 

*8.  V371I-  52.  v/|i.  56.  V8T^. 

57.  What  is  the  square  root  of  J  to  the  nearest  thou- 
sandth ? 

58.  What  is  the  square  root  of  W  to  the  nearest  thou- 
sandth ? 

59.  A  general  has  an  army  of  226576  men.  How  many 
must  he  place  rank  and  file  to  form  them  into  a  square  ? 

60.  A  gentleman  has  3  fields,  one  containing  3  acres  1 
square  -rod,  another  5  acres  69  square  rods,  and  the  third  6 
acres  91  square  rods.  He  wishes  to  excliange  them  for  a 
square  field  of  equal  area.     Find  the  side  of  the  square  field  ? 


INVOLUTION   AND   EVOLUTION.  249 

CUBE    ROOT. 

377.  To  extract  the  Cube  Root  of  a  number  is  to  find  one 
of  the  three  equal  factors  which  produce  it. 

378.  The  cube  of  a  number  contains  three  times  as  many 
figures  as  the  root,  or  three  times  as  many  less  one  or  tvjo. 
Thus, 

13  =:  1.  103  =  l/QOO.  1003  :^  I'OOO'OOO. 

43  =  64.  253  .=  15^625.  2433  =  14'348'907. 

93  =  729.         993  =  970^299.         9993  =  997'002'999. 

379.  If  a  third  power  is  separated  into  periods  of  three 
figures  each,  beginning  with  the  decimal  point,  the  number  of 
periods  will  show  the  number  of  figures  in  the  root.     Thus, 

The  cube  root  of  4'080'659.a92  contains  one  decimal 
and  three  integral  figures. 

380.  The  cube  of  the  highest  order  of  units  in  the  root  is 
found  in  the  highest  period  of  the  poiver.     Thus, 

9  tens3,  or  993  =  729  thousands,  or  729^000. 

9  hundreds3,  or  9003  ^  729  millions,  or  729'000'000. 

381.  Taking  any  number  composed  of  tens  and  ones,  as 
36,  separate  it  into  its  tens  and  ones,  cube  it,  keeping  the 
products  distinct,  and  we  have 

36  =:  30+6 

_36=  30+6 

216  =  (30  X  6)   +62 

108    ^  802  _^     (30  X  6) 


1296  =  302  +  2  (80  x  6)   +  6^ 

_36=  30+6 

7776  =  (302  X  6)  +  2  (30  X  62)  +  63 

3888    =303  +  2  (302  x  6)  +      (80  x  62) 

46656  =  303  ^  3  (302  x  6)  +  8  (30  x  62)  +  63 


250  INVOLUTION   AND    EVOLUTION. 

That  is,  the  cube  of  any  number  that  can  be  separated 
into  tens  and  ones  equals 

382.  The  cube  of  the  tens,  plus  three  times  the  product  of 
the  square  of  the  tens  and  the  ones,  plus  three  times  the  pro- 
duct of  the  tens  and  the  square  of  the  ones,  plus  the  cute  of 
the  ones. 

This  statement  may  be  represented  by  the  following 
formula  in  which  initial  letters  are  used,  and  the  sign  of 
multiplication  is  the  period. 

t3  +  3  t2 .  0  +  3  t .  o2  +  o3. 


WRITTEN    EXERCISES. 

61.   Eind  the  cube  root  of  405224. 

t3  +  3  t2.  o  +  3  t .  o2  +  o3  =    405^224  (70  +  4 
t3  =  343  000 

3t2=:702x3        ==14700 
3  t .  o  =  70  X  3  X  4  =      840 
o2  =  42  =.        16 


3  t2  +  3 1 .  o  +  o2    =:  15556 


62224  =:  3  t2 .  o  +  3t .  o2  +  o3 


62224  z=  (3 12  4-  3 1 .  o  +  o2)  X  o 


Solution.  —  This  power  has  two  periods  ;  hence  its  root  has  two 
figures,  tens  and  ones  (Art.  379).  The  cube  of  the  tens  is  in  the 
highest  period  (Art.  380).  The  greatest  tens^  in  405  thousands  is 
343000,  which,  subtracted,  and  its  cube  root  placed  at  the  right,  leaves 
62224,  which  equals  "  3  t^  X  o  +  3  t  X  o*  +  o^."  62224  consists 
principally  of  3  t*  X  o ;  hence,  if  we  divide  it  by  3  t*,  or  70'  X  3  = 
14700,  we  shall  have  the  ones,  which  we  find  to  be  4.  Finding  3  t 
X  o,  or  70  X  3  X  4  =  840,  and  o^,  or  4^  =  16,  and  adding  them  to 
14700,  we  have  14700  +  840  +  16  =  15556,  which  ecpials  3  t'  +  3  t 
X  o  +  o*.  Multiplying  this  by  the  ones,  4,  we  have  62224,  or  3  t*  X 
o  +  3  t  X  o*  +  o'.  Hence  we  conclude  that  74  is  the  cube  root 
required. 


INVOLUTION   AND    EVOLUTION. 


251 


Geometrical  Explanation  of  Cube  Root 

383.  As  the  length  of  a  cube  is  the  cube  root  of  its  con- 
tents, the  method  of  extracting  the  cube  root  of  a  number 
may  be  illustrated  by  the  process  of  ascertaining  the  length 
of  a  cube,  its  contents  being  given. 

62.  It  is  required  to  find  the  length  of  a  cube  containing 
13824  cu.  in. 

The  length  of  a  cube  containing  13824  cu.  in.  cannot  be  as  much 
as  30  in.,  for  a  30-inch  cube  contains  27000  cu.  in.  It  must  be  more 
than  20  in.,  for  a  20-inch  cube  contains  only  8000  cu.  in.  The  length 
of  the  given  cube,  then,  must  be  between  20  and  30  inches. 


Removing  from  the  given  cube,  A,  a  20-inch  cube,  5,  containing 
8000  cu.  in.,  there  remains  a  solid  containing  5824  cu.  in.  An  in- 
spection of  this  solid  shows  that  it  is 
largely  made  up  of  three  rectangular 
solids,  C,  D,  E,  having  square  faces 
corresponding  in  area  to  the  faces  of 
the  cube  removed.  It  is  evident  that 
the  thickness  of  these  solids  added  to 
the  length  of  the  cube  removed  will  be 
the  length  of  the  given  cube.  Now  the 
thickness  of  a  rectangular  solid  is  found 
by  dividing  its  contents  by  the  area  of 
its  face  (Art.  224).     We  find  the  area  of  one  of  the  square  faces  of 


252 


INVOLUTION   AND    EVOLUTION. 


each  of  these  rectangular  solids  to  be  20*,  or  400  sq.  in.,  and  the 
area  of  their  combined  square  faces  to  be  20^  X  3,  or  1200  sq.  in. 

Dividing  the  approximate  contents  of 
these  solids,  5824  cu.  in.,  by  this  area 
of  their  square  faces,  1200  sq.  in.,  we 
have  as  their  probable  thickness  4 
inches. 

Removing  the  three  rectangular 
solids,  C,  Dy  E,  there  remain  three 
smaller  rectangular  solids,  Fy  G,  H, 
whose  length  is  evidently  that  of  the 
cube  B  removed,  and  whose  width  is 
the  thickness  of  the  larger  rectangular  solids.     The  area  of  one  face 

of  one  of  these  solids  is  20  X  4,  and 
the  area  of  one  face  of  all  of  them  is 
20  X  4  X  3,  or  240  square  inches. 

Eemoving  these  three  smaller  solids, 
there  remains  the  little  cube  J,  whose 
face  corresponds  in  area  to  the  end  of 
one  of  the  smaller  rectangular  solids  re- 
moved, or  4*,  or  16  square  inches. 

Combining  the  area  of  the  faces  of 
the  six  rectangular  solids  with  that 
of  a  face  of  the  little  cube,  we  have  a  total  area  of  1200  +  240  + 
16,  or  1456  sq.  in.     Multiplying  the  area  of  one  face  of  these  seven 


F 

---'" 

-'-' 

i 

>< 

> 

F  GH 


4.00  80  80  80 


solids  by  their  thickness,  4,  we  have  4  X  1456,  or  5824  cu.  in.,  which 
is  the  number  of  cubic  inches  remaining  of  the  original  cube,  yl,  after 
the  removal  of  the  cube  B. 
the  given  cube  is  24  inches. 


We  therefore  conclude  that  the  length  of 


INVOLUTION    AND    EVOLUTION. 

The  work  is  expressed  thus  : 


253 


Contents.  Length. 

'  13^824  cu.  in.  (20  in.  +  4  in.  = 

Area.  8000         "  [24  in. 

202  X  3  =  1200  sq.  in. 


20  X  4    X  3  =    240 
42=      16 


1456 


5824 


5824 


63.    What  is  the  cube  root  of  14348907  ? 


14^348^907  (248 

23r=8 


202  X  3  =  1200 

20  X  4  X  3  =     240 

42=:       16 

1456 


2402  X  3  =  172800 

240  X  3  X  3  =      2160 

32=  9 


174969 


6348 


5824 


524907 


524907 


384.     Rule  for  finding  the  Cube  Root  of  a  Number. 

Beginning  at  the  decimal  point,  separate  the  given  power 
into  periods  of  three  figures  each. 

Find  the  greatest  cube  in  the  left  period,  and  place  its  root 
at  the  right.  Subtract  the  cube  of  this  root  from  the  left 
jyeriod,  and  to  the  remainder  annex  the  next  period  for  a  div- 
idend. 

Annex  a  cipher  to  the  root  already  found,  and  take  three 
times  its  square  for  a  trial  divisor.  Divide  the  dividend  by 
this  trial  divisor,  and  place  the  quotient  as  the  next  root 
figure. 


254 


INVOLUTION   AND    EVOLUTION. 


Multiply  the  number  last  squared  by  the  last  root  figure, 
and  add  three  times  the  product  and  the  square  of  the  last 
root  figure  to  the  trial  divisor  for  a  complete  divisor. 

Multiply  the  complete  divisor  by  the  last  root  figure,  sub- 
tract the  product  from  the  dividend,  and  to  the  remainder 
annex  a  new  period. 

Form  a  second  trial  divisor,  and  proceed  as  before  until  all 
tlie  periods  have  been  used. 

Note.  —  Note  2,  under  the  rule  for  square  root,  applies  likewise  to  cube 
root. 

Extract  the  cube  root  of  the  following  numbers  : 

64.   91125.  69.   12977875. 

70.  60236.288. 


65.  421875. 

66.  571787. 

67.  912.673. 

68.  3796416. 


71.  101847563. 

72.  258474853. 

73.  6372.783864. 


74.   What  is  the  cube  root  of  8.144865728  ? 
8.a44'865'728'  (2.012 


6 

! 

120000 

600 

1 

144865 

120601 

120601 

12120300 

12060 

4 

24264728 

1213236 

4 

24264728 

Here,  as  0  occurs  in  the  root, 
we  annex  00  to  the  trial  divi- 
sor, 1200,  and  bring  down  to 
the  corresponding  dividend 
another  period. 


75.  Find  the  cube  root  of  64481.201. 

76.  Find  the  cube  root  of  37259704. 

77.  What  is  the  cube  root  of  0.000001728  ? 

78.  What  is  the  cube  root  of  1860867  ? 

79.  What  is  the  value  of  V^8l44865728  ? 


INVOLUTION   AND   EVOLUTION.  255 


80.  What  is  the  value  of  v/0.075686967  ? 

81.  Find  the  value  of  v^O.008649  to  the  nearest  thousandth. 

82.  Find  the  cube  root  of  0.000007  to  the  nearest  thou- 
sandth. 

83.  What  is  the  cube  root  of  25  to  the  nearest  hundredth  ? 

385.  When  both  terms  of  a  fraction  are  perfect  cubes, 
the  cube  root  may  be  found  by  taking  the  cube  root  of 
each  term ;  but,  if  not,  reduce  the  fraction  to  a  decimal, 
and  then  find  the  root.  Mixed  numbers  may  be  changed 
either  to  improper  fractions  or  to  mixed  decimals. 


84.   What  is  the  cube  root  of  4^o¥^? 
.W"729       V^729" 

Solution.  1  /  JTTTTTT  =  ^, = 

V  4096      ^4096 


85.  What  is  the  cube  root  of  UMh  ^ 

86.  What  is  the  cube  root  of  49/y  ? 

87.  What  is  the  cube  root  of  J  ? 

88.  What  is  the  cube  root  of  -Sj  to  the  nearest  hundredth  ? 

89.  Find  the  cube  root  of  81 /y  to  the  nearest  hundredth. 

90.  What  is  the  cube  root  of  166|  ? 

91.  A  bushel  contains  2150.42  cubic  inches.  What  is  the 
depth  in  inches,  to  the  nearest  hundredth,  of  a  cubical  bin 
which  shall  contain  8  bushels  ?. 

QUESTIONS. 

360.  What  is  a  power  'I  The  first  power  ?  The  second  power  ? 
The  third  power?  361.  The  exponent  of  a  power?  362.  What  is 
involution  ? 

364.  What  is  a  root  ?  The  second,  or  square,  root  ?  The  third,  or 
cube,  root  ?  The  fourth  root  ?  365.  A  perfect  power  ?  367.  What 
is  evolution  ? 

369.  The  square  of  a  number  contains  how  many  times  as  many 
figures  as  its  root  ?  378.  The  cube  of  a  number  contains  how  many 
times  as  many  figures  as  its  root  ? 


256 


MENSURATION. 


MENSUBATION. 

386.    Mensuration  treats  of  the  measurement  of  lines, 
surfaces,  and  solids  or  volumes. 


RIGHT-ANGLED    TRIANGLES. 

387.  A  Right-angled  Triangle  has  one  right 
angle. 

The  Hypothenuse  is  the  side  opposite  the 
right  angle,  and  the  Perpendicular  is  the  side 
perpendicular  to  the  base. 

A  Right-angled 
Triangle. 

388.  It  will  be  seen  from  the 
diagram  that 

The  square  of  the  hypothenuse 
is  equal  to  the  sum  of  the  squares 
of  the  other  two  sides.   .  Hence, 

389.  To  find  the  hypothenuse, 

Take  the  square  root  of  the  sum 
of  tlie  squares  of  the  other  tioo 
sides. 

390.    To  find  the  base  or  perpendicular, 
Take  the  square  root  of  the  difference  of  the  squares  of  lln 
hypotlienuse  and  the  other  side. 

1.    The  base  of  a  right-angled  triangle  is  8,  and  the  pcrpen 
Jicular  6.     What  is  the  liypothenuse  ? 

Solution,  —  S'  +  6^  =  64  +  30  =  100 ;  V^lOO  =  10. 


MENSURATION.  257 

2.  The  hypothenuse  is  30,  and  one  of  the  sides  is  18.  What 
is  the  other  side  ? 

Solution.  —  302  _  ;^32  ^  900  _  324  ^  57(5 .  ^^^57^  ^  24 

3.  What  must  be  the  height  of  a  ladder  to  reach  to  the  top 
of  a  house  20  feet  high,  the  bottom  of  the  ladder  being  placed 
15  feet  from  the  base  of  the  house  ? 

4.  The  hypothenuse  is  157  feet,  and  the  perpendicular  132. 
What  is  the  other  side  ? 

5.  Two  vessels  sail  from  the  same  port ;  one  sails  due  south 
48  miles  and  the  other  due  west  36  miles.  What  is  then 
their  distance  from  each  other  ? 

6.  A  tree  stands  upon  the  edge  of  a  river  100  feet  wide,  and 
a  line  extending  from  the  opposite  shore  of  the  river  to  the 
top  of  the  tree  is  400  feet.  What  is  the  height  of  the  tree,  to 
the  nearest  hundredth  of  a  foot  ? 

7.  A  park  in  the  form  of  a  rectangle  is  40  rods  long  and  36 
rods  wide.  What  is  the  length  in  rods  of  a  walk  between  its 
opposite  corners  ? 

8.  A  ladder  32  feet  long  was  so  placed  in  a  street  as  to 
reach  a  window  25  feet  from  the  ground,  and  when  it  was 
turned  to  the  other  side,  without  changing  the  position  of  its 
foot,  it  reached  a  window  20  feet  from  the  ground.  How  wide 
was  the  street  ? 

QUADRILATERALS. 

391.  A  ftuadrilateral  is  a  plane  figure  bounded  by  four 
straight  lines. 


392.    Parallel  Lines  are  lines  in  the 
same  plane  having  the  same  direction. 


Parallel  Lines. 


393.    A  Parallelogram  is  a  quadrilateral  having  its  op- 
posite sides  parallel. 

17 


258 


MENSURATION. 


■liiii^ 


A  Rhomboid. 


A 


A  Rhombus. 


A  Eedangle  (Art.  166)  is  a  right-angled  parallelogram  ; 
a  RJwvihoid  is  a  parallelogram  having  no  right  angles ;  and 
a  Rhombus  is  a  rhomboid  having  equal  sides. 

394.  It  will  be  seen  by  the  dia- 
gram that  the  rhombus  A  B  C  D  is 
equal  to  the  rectangle  EBC F  of  the 
same  base  and  altitude  (Art.  218). 
Hence, 


S  A  F  D 

The  area  of  a  parallelogram  is  equal  to  the  product  of  the 
base  and  altitude. 

9.  What  is  the  area  of  a  parallelogram  whose  base  is  36  feet 
and  altitude  15  feet  ? 

10.  The  base  of  a  rhombus  is  16  feet  and  its  height  12  feet. 
What  is  its  area  ? 

11.  What  is  the  difference  in  the  area  of  two  floors,  the 
one  being  37  feet  long  and  27  feet  wide  and  the  other  40  feet 
long  and  20  feet  wide  ? 

395.  A  Trapezoid   is   a  quadrilateral 
aving  only  two  of  its  sides  parallel. 

396.  The  area  of  a  trapezoid  is  equal 
A  Trapezoid.        Iq  //^^  pvoduct  of  half  the  sum  of  tlie  par- 
allel sides  and  the  altit^cde. 

12.  What  is  tlie  area  of  a  trapezoid,  the  longer  of  the  two 
parallel  sides  being  120  feet,  the  shorter  100  feet,  and  the 
altitude  85  feet  ? 


MENSURATION.  259 

13.  What  is  the  area  of  a  plank  whose  length  is  6  meters, 
the  width  of  one  of  the  parallel  ends  being  60  centimeters 
and  the  other  40  centimeters  ? 

14.  The  parallel  sides  of  a  field  are  131  and  243  yards,  and 
the  breadth  220  yards.     How  many  acres  does  it  contain  ? 

397.  A  Trapezium  is  a  quadri- 
lateral having  no  two  of-  its  sides 
parallel. 

A  Diagonal  is  a  straight  line  join- 
ing any  two  angles  of  a  plane  figure 
not  adjacent,  as  the  line  A  C. 

A  Trapezium. 

398.  It  will  be  seen  from  the  above  diagram  that  a 
diagonal  divides  a  trapezium  into  two  triangles.     Hence, 

The  area  of  a  trapezium  is  equal  to  the  product  of  the  di- 
agonal and  half  the  sum  of  the  perpendiculars  drawn  to  the 
diagonal  from  the  vertices  of  opposite  angles. 

Note  1.  —  Any  plane  figure  bounded  by  straight  lines  is  called  a  Polygon, 
and  may  be  divided  into  triangles  ;  and  the  sum  of  the  areas  of  the  triangles 
will  be  the  area  of  the  figure. 

Note  2.  —  For  the  Circle  see  Art.  221. 

15.  The  diagonal  of  a  trapezium  is  16  feet,  and  the  perpen- 
diculars upon  it  from  the  opposite  angles  are  7  feet  and  5  feet. 
Find  the  area. 

7  ft.  +  5  ft. 
Solution.  —  ' X  16  =  96  sq.  ft. 

16.  What  is  the  area  of  a  trapezium  whose  diagonal  is  Q>b 
feet,  and  the  length  of  the  perpendiculars  let  fall  upon  it  from 
opposite  angles  is  14  feet  and  18  feet  ? 

17.  How  many  square  yards  of  paving  are  there  in  a  trape- 
zium whose  diagonal  is  found  to  measure  126  feet  3  inches, 
and  the  perpendiculars  upon  it  58  feet  6  inches  and  0^^  feet 
9  inches  ? 


260 


MENSURATION. 


PRISMS. 

399.  A  Prism  is  a  body  having  two  equal  parallej  poly- 
gons as  bases  and  the  other  faces  parallelograms. 

A  prism  is  triangular,  qiiadrangidar,  pentagonal,  etc., 

according  as  its  bases  have 
three  sides,  four  sides,  Jive  sides, 
etc. 

400.  The  contents  of  a-  jprism 
are  equal  to  the  product  of  the 
area  of  the  hase  hi/  the  altitude 

A  Quadrangular  ''  *^ 

Prism.        Qrp  length. 

Note.  —  For  the  Cylinder  see  Art.  225. 

18.  What  are  the  contents  of  a  triangular  prism  whose 
length  is  15  feet  and  the  area  of  its  triangular  base  is  6 
square  feet  ? 

19.  What  are  the  contents  of  a  quadrangular  prism  whose 
length  is  6  meters,  and  wbose  base  is  18  by  20  centimeters  ? 

20.  The  altitude  of  a  pentagonal  prism  is  20  feet  6  inclies, 
and  the  area  of  its  base  1075.30  square  inches.  What  are  its 
contents  in  cubic  feet  ? 


PYRAMIDS    AND    CONES. 

401.  A  Pyramid  is  a  body  whose  base 
is  any  polygon,  and  whose  sides  are  trian- 
gles meeting  at  a  point  called  the  vertex  of 
the  pyramid. 

A  pyramid,  like  a  prism,  is  triangular, 
quadrangular,    ^pentagonal,   etc., 
to  the  form  of  its  base. 


according 


A  Pyramid. 


MENSUKATION. 


26] 


402.  A  Cone  is  a  body  whose  base  is 
a  circle,  and  whose  convex  surface  tapers 
uniformly  to  a  point  called  the  vertex  of 
the  cone. 

The  altitude  of  a  pyramid  or  cone  is 
the  shortest  distance  from  the  vertex  to 
the  center  of  the  base ;  as  ^  ^. 

The  slant  height  is  the  shortest  dis- 
tance from  the  vertex  to  the  perimeter 
of  the  base ;  as  ^  (7. 


A  Cone. 


403.  The  contents  of  a  pyramid  or  cone  are  equal  to  the 
product  of  its  base  by  one  third  of  the  altitude.  Its  convex 
surface  equals  the  'product  of  the  perimeter  of  the  base  by 
half  the  slant  height, 

21.  What  are  the  contents  of  a  pyramid  whose  altitude  is  14 
feet  3  inches,  and  the  area  of  whose  base  is  14.70  square  feet  ? 

22.  What  are  the  contents  of  a  cone  whose  altitude  is  15.00 
meters,  the  circumference  of  the  base  being  12.5  meters  ? 

23.  The  largest  of  the  Egyptian  pyramids  is  square  at  its 
base,  and  measures  693  feet  on  a  side.  Suppose  its  other  sides 
to  meet  at  a  point  500  feet  above  the  base.  What  are  the 
contents  of  the  pyramid  in  cubic  feet  ? 

404.  A  Frustum   of   a 

pyramid  or  a  cone  is  the 
part  which  remains  after 
cutting  off  the  top  by  a 
plane  parallel  to  the  base. 

405.  The  contents  of  the 
frustum  of  a  pyramid  or 

cone  are  equal  to  the  sum  of  the  areas  of  the  two  bases  plus 
the  square  root  of  their  product,  multiplied  by  a  third  of  the 
altitude. 


262  MENSUKATION. 

24.  What  are  the  contents  of  the  frustum  of  a  square  pyra- 
mid whose  altitude  is  30  feet,  and  whose  side  at  the  base  is  20 
feet  and  at  the  top  10  feet  ? 

Solution, 
20  X  20  :=  400 ;  10  X  10  =  100  j  400  X  100  =  40000. 
V40000  =  200 ;  200  +  400  +  100  =  700. 
30  4-  3  ==  10 ;  700  X  10  =  7000  cu.  ft. 

25.  What  are  the  contents  of  a  column  whose  altitude  is  2S 
feet  6  inches,  and  whose  diameter  at  the  larger  end  is  3  feet  and 
at  the  other  2  feet  6  inches  ? 

26.  How  many  cubic  feet  in  a  cquare  stick  of  timber  whose 
length  is  18  feet  8  inches,  and  whose  side  at  the  larger  end  is 
27  inches  and  at  the  smaller  is  16  inches  ? 

406.  A  Sphere  is  a  body  bound- 
ed by  a  curved  surface,  all  parts  of 
which  are  equally  distant  from  a 
point  within  called  the  center. 

The  Diameter  of  a  sphere  is  any 
straight  line  drawn  through  its 
center,  and  terminating  both  ways 
in  the  surface;  and  the  Circum- 
ference is  the  greatest  distance 
around  the  sphere. 


A  Sphere. 


407.  The  surface  of  a  sphere  is  equal  to  the  product  of 
3.1416  hy  the  square  of  the  diameter  ;  and 

The  contents  of  a  sphere  are  equal  to  the  product  of  ^  of 
3.1416  hy  the  cidte  of  the  diameter. 

27.  What  is  the  surface  of  a  sphere  whose  diameter  is 
26  inches  ? 

28.  What  number  of  square  meters  of  gold-leaf  will  gild 
a  globe  18  centimeters  in  diameter  ? 


MENSURATION.  263 

29.  How  many  cubic  feet  of  gas  will  be  required  to  fill  a 
spherical  balloon  whose  diameter  is  50  feet  ? 

30.  How  many  cubic  miles  of  volume  has  the  moon,  allowing 
its  diameter  to  be  2100  miles  ? 

SIMILAR   SURFACES. 

408.  Similar  Surfaces  are  surfaces  having  the  same 
form,  without  regard  to  size. 

409.  Similar  surfaces  are  to  each  other  as  the  squares  of 
their  correspoiiding  dimensions.     Hence, 

The  corresponding  dimensions  of  similar  surfaces  are  to 
each  other  as  the  square  roots  of  their  areas. 

31.  A  triangle  whose  base  is  12  feet  has  an  area  of  72 
square  feet.  What  is  the  base  of  a  similar  triangle  whose 
area  is  32  square  feet  ? 

Solution,  — .  72  :  32  =:  12^  :  x%  or  72  :  32  =  12*:  8.*   Ans.  8ft. 

32.  A  side  of  one  of  two  similar  triangles  is  12  feet,  and 
the  corresponding  side  of  the  other  is  8  feet.  If  the  larger  of 
the  triangles  has  an  area  of  72  square  feet,  what  is  the  area 
of  the  other  ? 

33.  The  areas  of  two  similar  rectangular  fields  are,  respec- 
tively, 9  and  6.25  acres,  and  the  first  is  120  rods  long.  What 
is  the  length  of  the  second  ? 

34.  The  diameters  of  two  circles  are,  respectively,  100  feet 
and  50  feet.     How  much  larger  is  the  one  than  the  other  ? 

35.  If  a  pipe  whose  diameter  is  1.5  inches  fills  a  cistern  in  5 
hours,  in  what  time  will  another  whose  diameter  is  35  inches 
fill  it  ? 

36.  If  it  costs  $  30  to  pave  the  bottom  of  a  cellar  whose 
width  is  16  feet,  what  will  it  cost  to  pave  a  similar  one  whose 
width  is  24  feet? 


264  MENSURATION. 

37.  A  gentleman  has  a  park,  in  the  form  of  a  right-angled 
ijiangle,  containing  105.55  square  meters,  the  longest  side  of 
which  is  15  meters.  He  wishes  to  lay  out  another  in  the 
same  form,  whose  longest  side  shall  be  3  times  the  length  of 
the  first.     Required  the  area. 

SIMILAR    SOLIDS. 

410.  Similar  Solids  are  solids  having  the  same  form 
without  regard  to  size. 

411.  Similar  solids  are  to  each  other  as  the  cubes  of  their 
corresponding  dimensions.     Hence, 

412.  The  corresponding  dimensions  of  similar  solids  are 
to  each  other  as  the  cube  root  of  their  contents  or  volumes. 

38.  If  a  cone  3  feet  in  height  contains  1539  cubic  feet,  what 
are  the  contents  of  a  similar  cone  2  feet  in  height  ? 

Solution.  —  33  :  23  =  1539  :  a^,  or  27  :  8  ==  1539  :  456. 

39.  Two  similar  bins  contain,  respectively,  400  and  600 
bushels.     If  the  first  is  4  feet  deep,  how  deep  is  the  second  ? 

40.  A  sugar  loaf  which  is  12  inches  high  weighs  16  pounds. 
How  many  inches  may  be  broken  from  the  base,  that  the  rest 
msij  weigh  8  pounds  ? 

41.  If  a  sphere  6  inches  in  diameter  weighs  16.50  kilo- 
grams, what  is  the  weight  of  a  sphere  of  the  same  material  12 
inches  in  diameter  ? 

42.  Suppose  the  diameters  of  the  earth  and  the  moon  to  be, 
respectively,  8000  and  2000  miles,  how  many  times  larger  is 
the  earth  than  the  moon  ? 

43.  If  the  weight  of  a  well-proportioned  man  5  ft.  6  in.  in 
height,  is  140  pounds,  what  should  be  the  weight  of  Bar- 
num's  Chinese  giant  who  is  8  ft.  3  in.  in  height  ? 


REVIEW.  265 


REVIEW. 


ORAL    EXERCISES. 

413.  1.  If  6  barrels  of  flour  cost  $  33,  what  will  11  bar- 
rels  cost  ? 

2.  If  9  tons  of  coal  cost  $  54,  bow  many  cords  of  wood  at3 

I  4  a  cord  would  cost  as  much  as  5  tons  of  coal  ? 

3.  If  6  horses  in  a  certain  time  consume  14  bushels  ot 
oats,  how  many  bushels  will  10  horses  consume  in  the  same 
time? 

4.  If  I  of  a  barrel  costs  $  4§,  what  will  f  of  a  barrel  cost  ? 

5.  If  A  can  do  a  piece  of  work  in  3  days,  and  B  can  do  the 
same  in  4  days,  what  part  of  the  work  can  they  together  do  ii* 
Iday? 

6.  If  A  and  B  can  in  1  day  perform  ^  of  a  piece  of  work, 
how  long  will  it  take  them  to  do  the  whole  ? 

7.  Divide  75  into  two  numbers  that  shall  be  to  each  other 
as  7  to  8  ? 

8.  The  first  three  terms  of  a  proportion  are  7,  8,  and  9. 
What  is  the  fourth  term  ? 

9.  How  long  will  40  bushels  of  oats  last  3  horses,  if  they 
consume  8  bushels  in  2  weeks  ? 

10.  If  7  horses  consume  3  tons  of  hay  in  6  weeks,  how 
many  tons  will  6  horses  consume  in  4  weeks  ? 

11.  A  and  B  run  a  race  in  which  A  gains  3  rods  in  running 

II  rods.     How  far  must  he  run  to  gain  7  rods  ? 

12.  A  box  of  tea  lasted  a  man  and  his  wife  9  months. 
When  the  man  was  absent  it  would  last  his  wife  12  months. 
How  long  would  it  have  lasted  the  man  alone  ? 

13.  Smith  and  Jones  hire  a  pasture  together.  Smith  pays 
$  16  and  Jones  $  20.  Smith  puts  in  12  sheep.  How  many 
does  Jones  put  in  ? 

14.  A  furnished  2  loaves  for  a  lunch,  and  B  furnished 
3,  while  C  contributed  30  cents  to  be  divided  between  A  and 
B.     How  much  should  each  receive  ? 


266  REVIEW 

15.  Three  men  trade  together.  A  puts  in  $  3  as  often  as  B 
puts  in  $  4  and  as  often  as  C  puts  in  $  5.  They  gain  1 87. 
What  was  each  man's  share  of  the  gain  ? 

16.  A  hare  is  60  rods  before  a  hound,  and  runs  3  rods  while 
the  hound  runs  6.  How  many  rods  must  the  hound  run  to 
overtake  the  hare  ? 

17.  What  is  the  time  of  day  when  ^  of  the  time  past  mid 
aight  equals  the  time  to  noon  ? 

18.  At  what  time  between  4  and  5  o'clock  are  the  hour  and 
the  minute  hand  together  ? 

19.  A  can  do  a  piece  of  work  in  2  days,  B  can  do  it  in  4 
days,  and  C  in  3  days.    In  what  time  can  they  do  it  together  ? 

20.  If  I  pay  $  15  four  months  before  it  is  due,  how  long 
after  it  is  due  may  I  keep  $  20  to  balance  it  ? 

21.  If  24  men  can  mow  66  acres  of  grass  in  2  days,  how 
many  acres  can  14  men  mow  in  7  days  ? 

22.  What  is  the  interest  of  1 800  for  36  days  at  10  %  ? 

23.  I  bought  12  cords  of  wood  that  was  to  have  be^n  4  ft. 
long.  It  proved,  however,  to  be  only  42  in.  What  was  my 
per  cent  of  loss  ? 

24.  William  Simonds  receives  a  bequest  of  $840,  with 
which  he  buys  U.  S.  4  per  cents  at  120.  What  is  his  annual 
income  ? 

25.  In  what  time  will  $  400  yield  $  72  interest  at  4  %  ? 

26.  What  is  the  perimeter  of  a  square  10-acre  field  ? 

27.  My  semi-annual  income  from  an  8  %  stock  is  $  84. 
How  many  shares  do  I  own  ? 

28.  What  is  the  longest  straight  line  that  can  be  drawn  on 
a  sheet  of  6  X  8  paper  ? 

29.  A  lawyer  earns  $  275  collecting  money  at  5  %  commis- 
sion.    What  amount  does  he  collect  ? 

30.  The  diagonal  of  a  garden  8  rods  long  is  10  rods.  Ee- 
quired  the  area  of  the  garden. 


REVIEW.  267 

31.  The  interest  of  1 400  is  $  70  for  2  y.  6  mo.  What  is 
the  rate  ? 

32.  What  is  the  cube  root  of  the  square  of  |  ? 

33.  Which  is  the  larger,  the  square  of  J  or  its  cube,  and 
how  much  ? 

34.  Find  a  mean  proportional  between  3  and  48. 

35.  What  is  the  inverse  ratio  of  g  to  }  ? 

36.  Received  a  discount  for  cash  of  12^  %  on  a  bill  of  goods 
amounting  to  $  128.     How  much  did  I  pay  ? 

37.  Which  is  the  better  investment,  6  %  stock  at  120  or  8  % 
bonds  at  144  ? 

38.  Find  the  missing  term  in  the  proportion  2^  :  x  =  5  :  7|. 

39.  A  note  for  $  800,  which  matures  May  31,  is  dis- 
counted April  1.     Required  its  proceeds. 

40.  In  selling  cloth  at  $  6  per  yard,  I  gain  20  fo.  What  % 
should  I  gain  if  I  sold  it  at  $  6.40  ? 

41.  A  bank  renewed  $  100000  worth  of  U.  S.  6  per  cents  at 
3 J  %.     What  is  the  annual  loss  of  interest  ? 

42.  What  is  the  interest  of  the  present  worth  of  a  debt  of 
$  520  due  in  a  year,  money  being  worth  4  %  ? 

43.  When  exchange  on  London  is  quoted  at  |  4.80,  what 
will  a  £  200  draft  cost  ? 

44.  I  lent  Willard  Aldrich  $  1500  for  three  months.  How 
long  should  he  lend  me  $  1000  to  requite  the  favor  ?    • 

45.  What  is  the  ratio  of  the  square  of  4  to  the  cube  root  of 
125? 

46.  What  is  the  inverse  ratio  of  a  decigram  to  a  kilogram  ? 

47.  A  party  of  pleasure-seekers  hired  a  sailboat  for  $  4,  each 
paying  as  many  cents  as  there  were  persons  in  the  party. 
What  did  each  pay  ? 

48.  A  man  with  more  money  than  learning  willed  -J  his 
property  to  his  wife,  J  to  his  son,  and  |  to  his  daughter.  He 
left  $  13000  ;  how  ought  it  to  be  equitably  divided  ? 


268  REVIEW. 

WRITTEN    EXERCISES. 

49.  I  am  offered  $  7000  cash  for  my  house  or  $  7700  in  a 
year  without  interest.  I  can  get  10  %  for  my  money.  Which 
is  the  better  offer  ? 

50.  I  buy  butter  for  32  cents,  and  am  obliged  to  sell  it  for 
28  cents.     What  %  do  I  lose  ? 

51.  A  suit  of  clothes  marked  to  sell  at  %  40  is  sold  at  10  % 
discount,  and  yet  the  seller  makes  25%.  What  did  it  cost 
him? 

52.  What  is  33^  %  of  the  square  of  the  cube  root  of  f  |  ? 

53.  What  is  the  entire  surface  of  a  14-foot  cube  ? 

54.  When  Washington  was  inaugurated  there  were  13 
States  in  the  Union,  and,  March  4,  1881,  there  were  38.  Re- 
quired  the  rate  per  cent  of  increase. 

55^  I  lost  16§  %  of  my  property  by  fire  and  20  %  by  failures. 
How  much  had  I  originally  if  %  5700  remained  ? 

56.  I  buy  books  at  80  cents,  20  %  off,  and  sell  for  $  1.00, 
15  %  off,  and  5  %  off  the  remainder.     How  much  do  I  gain  ? 

57.  Received  a  consignment  of  2000  barrels  of  flour,  which 
I  sell  at  *8.50,  paying  $74  storage  and  $27  cartage.  My 
commission  is  1^%.     How  much  do  I  remit  ? 

58.  My  insurance  at  J  %  cost  me  a  premium  of  %  62.40,  and 
is  I  of  the  value  of  my  house.  What  do  I  lose  in  case  of  its 
total  destruction  by  fire  ? 

59.  Received  a  remittance  of  $  25375  for  the  purchase  of 
cotton  at  12^/  per  pound  after  deducting  my  commission  of 
1|  %.     How  many  pounds  do  I  purchase  ? 

60.  I  buy  6  %  stock  at  112.  What  per  cent  do  I  receive 
on  my  investment? 

61.  What  sum  will  cancel  a  note  of  $892,  Dec.  24,  1881, 
dated  May  30,  1878,  drawing  4i  %  interest  ? 

62.  Sold  two  pianos  for  $600  each,  gaining  20%  on  one 
and  losing  20%  on  the  other.     Required  my  gain  or  loss. 


REVIEW.  269 

63.  What  is  the  compound  interest,  payable  Jan.  1  and 
July  1  of  each  year,  of  $  1200  at  a  yearly  rate  of  4  %  for  1  y. 
9  mo.  18  d.  ? 

64.  Eobert  Burns  owes  me  $  1500  due  9  mo.  hence  without 
interest.  What  will  be  a  fair  deduction  for  present  payment 
if  the  current  rate  for  money  is  8  %  ? 

«5.   How  long  will  $  1000  be  in  amounting  to  $  1500  at  7  %  ? 

66.  A  note  for  $  1600  dated  Jan.  1,  1879,  has  three  indorse- 
ments of  $  200  each,  —  July  10,  1879,  Aug.  15,  1880,  and 
May  12,  1881.     What  was  due  at  settlement,  Jan.  1,  1882  ? 

67.  Bought  62|  yards  of  carpet  at  $  1.87|,  receiving  a  dis- 
count of  15  %.     What  was  my  bill  ? 

68.  Sold  an  estate  which  cost  me  $  8000,  |  %  for  insurance, 
$  64  taxes,  and  1 1260  for  repairs,  for  a  2-month  note  of 
$  11000,  which  I  had  immediately  discounted  at  a  bank  at 
5  %.     Did  I  gain  or  lose,  and  how  much  ? 

69.  What  is  the  face  of  a  4-month  note  which  will  yield 
$  800  when  discounted  at  4  %  ? 

70.  What  cost  125  shares  of  N.  Y.  Central  E.  R.  stock  at 
143  i  and  brokerage  ? 

71.  May  14,  I  get  a  90-day  note  for  $  1292  discounted, 
which  was  dated  April  10,  1882.     What  were  the  proceeds  ? 

72.  Exchange  on  Paris  is  quoted  at  5.14.  What  will  a  bill 
of  exchange  for  8500  francs  cost  me  ? 

73.  Find  the  missing  term  : 

V^:0625  :  (1)2  =  X  :  y/ 15.625. 

74.  Find  a  mean  proportional  to  816  and  97  J. . 

75.  If  J  of  a  pound  cost  I  f ,  what  will  f  of  an  ounce  cost  ? 

76.  If  an  8-cent  loaf  weighs  10  ounces  with  flour  at  $  8.50 
per  barrel,  what  will  a  12-cent  loaf  weigh  with  flour  at  $  10  ? 

77.  If  my  gas-bill  is  $  4  for  the  month  of  January,  1881, 
when  I  use  4  burners  3  J  hours  each  evening,  what  will  it  be 
for  the  month  of  February,  when  I  use  3  burners  4  h.  20  min. 
each  evening  ? 


270  REVIEW. 

78.  Three  children,  aged  14,  12,  and  7  respectively,  share  a 
bequest  of  $  7000  in  proportion  to  their  ages.  What  is  the 
share  of  the  youngest  ? 

79.  Ames  and  Stevens  form  a  partnership  January  1,  Ames 
furnishing  $  8000  capital  and  Stevens  $  6000.  May  1st,  they 
take  Conant  into  the  firm  with  a  capital  of  $  5000,  and  Au- 
gust 1,  Hubbell  joins  them  with  $  3000.  How  shall  the  gain 
of  $  12000  be  divided  at  the  end  of  the  year  ? 

80.  If  it  costs  $320  to  fence  a  rectangular  lot  120  rods 
long  and  80  rods  wide,  what  will  it  cost  to  fence  an  equal 
square  lot  at  the  same  rate  ? 

81.  If  it  is  90  feet  between  the  bases  of  a  ball  ground, 
what  is  the  shortest  distance  from  the  second  base  to  the  home 
plate  ? 

82.  What  is  the  diagonal  of  a  section  of  land  in  rods  ? 

83.  How  long  a  cube  contains  25934.336  cu.  ft.  ? 

84  Grandmother  Gray  lives  in  the  center  of  a  square  farm 
containing  a  quarter-section  of  land.  How  far  must  she 
walk  by  the  shortest  route  to  visit  her  four  children,  one  of 
whom  lives  at  each  corner  of  the  farm,  and  return  home  ? 

85.  How  large  a  cubical  pile  will  32  cords  of  wood  make  ? 

86.  Eind  the  diagonal  of  a  rectangular  solid  60  inches  long, 
40  wide,  20  high. 

87.  How  many  feet  long  is  a  square  containing  an  acre  ? 

88.  What  will  it  cost  at  $  0.75  per  rod  to  fence  eight  equal 
rectangular  lots  made  from  a  square  ten-acre  field  ? 

89.  The  roof  of  how  high  a  building  can  be  reached  by 
a  40-foot  ladder,  the  bottom  of  which  is  12  feet  from  the 
building  ? 

90.  A  certain  cube  contains  175616  cu.  in.  Find  the  diag- 
onal  of  one  of  its  faces. 

91.  If  5184  is  the  square  of  a  number,  what  is  its  cube  ? 

92.  How  long  are  the  rafters  of  a  house  40  feet  wide,  the 
ridge-pole  being  12  feet  above  the  attic  floor  ? 

93.  What  is  the  ratio  of  V^  2|"  to  V^  4jf 


REVIEW.  271 

94.  What  is  the  entire  area  of  a  cube  containing  592704 
cubic  inches  ? 

95.  If  8  men  can  make  300  pairs  of  shoes  in  6  days,  how 
many  men  must  be  added  to  their  number  in  order  that  twice 
as  much  may  be  done  in  half  the  time  ? 

96.  There  are  two  numbers  in  the  ratio  of  6  to  11,  and  the ' 
larger  is  167  J^.     What  is  the  smaller  ? 

97.  Reduce  the  ratio  3^2  •  f  ^^  ^^^^  smallest  integral  terms. 

98.  If  4/g^  pounds  of  coffee  cost  $  1.38,  what  will  11 1  pounds 
cost  ? 

99.  If  24  men  can  build  405  yards  of  wall  in  24  days,  how 
many  men  will  it  require  to  build  it  in  8  days  ? 

100.  A  and  B  are  in  partnership  for  one  year.  January 
1,  A  put  in  $  2000,  but  B  did  not  put  in  any  until  April  1. 
What  did  he  then  put  in  to  have  an  equal  share  with  A  at 
the  end  of  the  year  ? 

101.  A  and  B  start  from  the  same  place,  and  travel  the 
same  road.  A  goes  5  days  before  B,  at  the  rate  of  20  miles 
a  day.  B  follows  at  the  rate  of  25  miles  a  day.  In  what 
time  and  at  what  distance  will  he  overtake  A  ? 

102.  A  cistern  has  3  pipes :  the  first  will  empty  it  in  2 
hours,  the  second  in  3  hours,  and  the  third  in  4  hours.  In 
what  time  would  they  together  empty  the  cistern  ? 

103.  A  certain  piece  of  labor  was  to  have  been  performed 
by  144  men  in  36  days,  but  a  number  of  them  having  been 
sent  away,  the  work  was  performed  in  48  days.  What  nun^* 
ber  of  men  was  se»t  away  ? 

104.  If  15  carpenters  can  build  a  bridge  in  60  days  by 
working  15  hours  a  day,  how  long  will  it  take  20  men  to 
build  the  bridge  by  working  only  10  hours  a  day  ? 

105.  Divide  $  2000  among  A,  B,  and  C,  so  that  for  every 
$  3  given  to  A,  B  is  to  receive  $  5,  and  C$8.  What  sum  did 
they  each  receive  ?  ^ 

106.  How  many  inches  in  the  diagonal  of  a  square  whose 
side  is  24  feet  ? 


272  KEVIEW. 

107.  What  is  the  cube  root  of  |,  to  the  nearest  hundredth  ? 

108.  The  side  of  a  square  is  8  feet  6  inches.  What  is  the 
side  of  a  square  having  25  times  the  area  ? 

109.  How  many  gallons  per  minute  will  a  |-inch  faucet 
discharge,  if  a  ^inch  faucet  discharges  30  gallons  ? 

110.  One  person  owes  another  $  150  payable  in  six  months, 
$  180  payable  in  8  months,  and  $  270  payable  in  4  months. 
Find  the  equated  time  of  payment. 

111.  Three  towns  are  to  provide,  according  to  their  popula- 
tion, a  contingent  of  182  men.  The  population  of  the  first 
town  is  2456,  of  the  second  735,  and  of  the  third  436.  Find 
as  exactly  as  possible  the  number  of  men  to  be  provided  by 
each  town. 

112.  How  many  square  yards  in  the  area  of  the  sides  of  a 
square  pyramid,  whose  slant  height  is  100  feet  and  the  perime- 
ter of  whose  base  is  54  feet  ? 

113.  How  many  globes  4  inches  in  diameter  are  equal  in 
volume  to  one  that  is  12  inches  in  diameter  ? 

REVIEW    QUESTIONS. 

134.  How  is  the  part  that  one  number  is  of  another  found  ?  338. 
What  is  ratio  ?  How  is  it  determined  ?  345.  What  is  proportion  1 
349.  What  is  a  simple  proportion  ?    351.  A  compound  proportion  ? 

161.  What  is  a  hne?  164.  What  is  an  angle?  165.  What  is  a 
right  angle  ?     179.  How  many  degrees  in  a  right  angle  ? 

163.  What  is  a  surface?  166.  What  is  a  rectangle?  167.  A 
square  1     391.  A  quadrilateral  ? 

219.  What  is  a  triangle?  387.  A  right-angled  triangle?  The 
hypothenuse  ?     The  perpendicular  ? 

177.  What  is  a  circle?  The  circumference  of  a  circle?  The  di- 
ameter ?     222.  The  ratio  of  the  cicumference  to  the  diameter  ? 

169.  What  is  a  solid,  or  volume  ?     223.  A  rectangular  solid  ? 

170.  A  cube?  399.  ^ prism?  225.  A  cylinder?  401.  A  pyra- 
mid 1    402.  A  cone  ?    406.  A  sphere  ? 


EXAMINATION   QUESTIONS.  273 


EXAMINATION    QUESTIONS. 

FOR  TESTING  PROFICIENCY,  FOR  PROMOTIONS,  AND  FOR  S1DB 
PLEMENTARY  PRACTICE.  ARRANGED  FROM  PAPERS  USED 
IN  VARIOUS  CITIES. 

FUNDAMENTAL  RULES. 

414.  1.  Write  in  words  the  following:  4238  —  758  = 
145  X  24. 

2.  What  is  the  difference  between  24  times  325  and  36 
times  245  ? 

3.  How  many  more  times  is  16  contained  in  192  than  it  is 
in  64? 

4.  Multiply  125  by  9,  and  write  the  product  in  figures  and 
in  words. 

6.  John  has  20  marbles  and  James  has  12.  How  many 
marbles  must  John  give  James  that  each  may  have  the  same 
number? 

6.  The  salary  of  the  President  of  the  United  States  is 
$50000  a  year.     How  much  is  that  a  month  ? 

7.  What  number  must  be  added  to  365  to  make  730? 

8.  Henry  is  Id  years  old,  and  one  half  of  his  age  is  twice 
the  age  of  his  brother.     What  is  his  brother's  age  ? 

9.  (12  X  9)  + 12  =  5  X? 

10.  How  many  hours  are  there  in  January  ? 

415.  1.  The  product  of  three  factors  is  56700;  two  of  the 
factors  are  42  and  75.     What  is  the  third  factor  ? 

2.  The  dividend  is  50000,  the  quotient  is  136,  and  the  re- 
mainder 360i     What  is  the  divisor? 

3.  Divide  149184  by  84;  and  write  the  quotient  in  figures 
and  in  words. 

4.  Two  men,  who  are  1224  miles  apart,  travel  towards  each 
other ;  one  32  miles  a  day  and  the  other  36  miles  a  day.  In 
how  wiauy  days  will  they  meet  ? 


274  EXAMINATION    QUESTIONS. 

5.  The  miimend  being  26402  and  the  remainder  18725, 
what  is  the  subtraliend  ? 

6.  The  product  is  how  many  times  the  multiplicand  ? 
When  is  it  a  concrete  number  ? 

7.  Multiply  40800  by  30600.  Why  is  the  product  an  ab^ 
fitract  number  ? 

Q.  A  farmer  bought  8  horses  for  $  75  each^  and  6  horses 
for  $  125  each,  and  sold  them  all  for  $  120  each.  How  many 
dollars  did  he  gain  ? 

9.   Divide  381600  by  123,  and  prove  your  work. 
10.    A  man  bought  240  acres  of  land  at  $  26  an  acre,  giving 
in  payment  a  house  valued  at  $  2820  and  horses  valued  at 
$  180  each.     How  many  horses  did  he  give  ? 

416.  1.  What  is  the  difference  between  a  figure  and  a 
number  ? 

2.  Kead  40090.049.  What  name  is  given  to  the  number  at 
the  left  of  the  point  ? 

3.  How  may  you  prove  subtraction  ?  Illustrate  by  an 
example. 

4.  Give  the  sum  of  all  the  numbers  in  the  next  four  ex- 
amples. 

5.  If  3008.7  is  the  minuend  and  299.99  is  the  subtrahend^ 
what  is  the  remainder  ? 

6.  If  8467  is  the  remainder  and  44  is  the  subtrahend,  what 
is  the  minuend  ? 

7.  Multiply  387.5  by  6.  Perform  the  same  example  by 
raddition. 

8.  Divide  86784  by  87,  and  prove  the  work. 

9.  A  man  bought  a  cow  for  $  85,  a  horse  for  .$  165,  and  a 
carriage  for  $  276.  How  much  more  did  he  pay  for  the  car- 
riage than  for  both  horse  and  cow  ? 

10.  A  man  sold  108  acres  of  land  at  $  205  per  acre,  and 
svith  the  money  purchased  horses  at  $75  each  ;  how  many  did 
he  ^(it  ? 


EXAMINATIO]^   QUESTIONS.  275 

417.  1.  The  multiplicand  is  87040,  the  multiplier  is  6080. 
What  is  the  product  ? 

2.  One  cord  of  wood  contains  128  cubic  feet.  How  many 
cubic  feet  are  there  in  75  cords  of  wood  ? 

3.  A  farmer  sold  to  a  flour  merchant  45  bbl.  of  apples  at 
$  3  per  bbl.,  65  bbl.  of  potatoes  at  $  2  per  bbl.,  and  receiyed 
in  payment  40  bbl.  of  flour  at  $  6  per  bbl.,  and  the  balance  in 
money.     How  many  dollars  did  he  receive  ? 

4.  If  a  silk  dress  containing  17  yards  costs  $  38.25,  what  is 
the  cost  a  yard  ? 

5.  84.61  X  27  =  ?    At  the  right  of  each  term  write  its  name. 

6.  How  many  pounds  of  sugar  at  12  cents  a  pound  can  you 
get  for  18  dozen  eggs  at  16  cents  a  dozen  ? 

7.  Multiply  814  by  16 ;  add  279  to  the  product ;  subtract 
384  from  the  sum,  and  divide  the  remainder  by  18. 

8.  Show  by  an  example  that  either  of  the  factors  in  multi- 
plication maybe  used  as  multiplier  without  changing  the  value 
of  the  product. 

9.  A  man  bought  8  cords  of  wood  at  $  6.50  per  cord,  18 
tons  of  hay  at  $  21  per  ton,  7  bushels  of  potatoes  at  $  0.90  per 
bushel.     He  paid  $  75  in  cash.    How  much  does  he  still  owe  ? 

10.  A  man  bought  ten  books ;  for  4  of  thorn  he  paid  1 1.50 
each,  for  3  of  them  he  paid  $  1.80  each,  and  for  the  rest  he 
paid  28  cents  each.     How  much  did  he  pay  for  all  ? 

418.  1.  Divide  the  product  of  the  sum  and  difference  of 
125  and  36  by  48. 

2.  Bought  360  acres  of  land  for  $  32400,  and  sold  it  for 
$  8400  more  than  cost.     What  was  the  selling  price  per  acre  ? 

3.  (i  of  69543248)  -  (^  of  81369)  =  ? 

4.  I  bought  1265  books,  and  sold  ^  of  them  at  $  0.50  each 
and  the  remainder  for  $  0.75  each.  How  much  did  I  receive 
for  them  ? 

5.  Bought  18  lb.  of  steak  at  24/,  41-  doz.  eggs  at  36/, 
3  qt.  of  molasses  at  18/,  and  a  bushel  of  potatoes  for  75/. 
Required  the  amount  of  my  bill. 


276  EXAMINATION   QUESTIONS. 

6.  In  a  certain  church  there  are  40  pews  that  seat  6  people, 
35  that  seat  5,  and  18  that  seat  4.  The  gallery  will  accom- 
modate 115.  At  a  lecture  every  seat  is  filled ;  the  price  of 
admission  being  25/,  what  were  the  proceeds,  62  compliment- 
ary tickets  having  been  used  ? 

7.  A  farmer  bought  a  cow  and  254  sheep  for  $  1134.50 
The  cow  cost  $  55.     What  did  54  sheep  cost  ? 

8.  How  many  5-cent  pieces  in  $  720  ? 

9.  Exchanged  a  farm  worth  $4278  for  75  sewing-machines 
worth  $  45  each,  and  $  200  cash.  Did  I  gain  or  lose,  and  how 
much  ? 

10.  6010  X  6020  X  9  =  18  X  ? 

COMMOISr    FRACTIONS. 

419.  1.  Subtract  the  sum  of  all  the  prime  numbers  from 
1  to  37  from  that  of  all  the  composite  numbers  from  4  to  40. 

2.  Name  three  composite  numbers  prime  to  each  other. 

3.  Keduce  2%  f  ?  i  h  ^^^  ^^  ^^  fractions  having  the  least 
common  denominator,  and  find  their  amount. 

4.  Reduce  to  their  lowest  terms  ^^^j,  }||,  ||f,  and  x^Vt, 
using  in  each  case  the  greatest  common  divisor. 

5.  What  is  2i  fractional  unit,  and  what  is  the  unit  of  a  frac- 
tion ?     Give  an  example  of  each. 

6.  Find  the  amount  of  the  following  mixed  numbers  :  4/^5, 
6/2'V,  and  12^*^^.  Reduce  to  lowest  terms  and  least  common 
denominator  before  you  add. 

7.  From  175 J  take  95^,  and  from  45  g  take  25f. 

8.  In  what  two  ways  can  fractions  be  multiplied  by  an  in- 
teger ?     Which  way  is  preferable  ?     Why  ? 

9.  Multiply  175 J  by  12,  124  by  6f ,  and  ^  by  12|. 

10.  A  man  who  owned  3  of  a  ship  sold  %  of  his  interest  for 
$30000.  What  part  of  the  sliip  did  he  sell,  and  what  was  the 
value  of  the  wliole  ship  at  that  rate  ? 


EXAMINATION   QUESTIONS.  277 

420.  1.  Define  a  prime  number ;  a  composite  number  ;  a 
fraction. 

2.  Find  the  greatest  common  divisor  of  182  and  196. 

3.  Find  the  least  common  multiple  of  8,  7,  10,  14. 

4.  Change  13J,  61|,  15^,  to  improper  fractions. 

5.  Keduce  f f f  and  |ff  to  their  smallest  terms,  using  the 
greatest  common  divisors. 

6.  Reduce  ^|§^,  ^{j-y  -?§-;  ^^  whole  or  mixed  numbers. 

7.  Eeduce  |,  f,  -f^,  f,  to  equivalent  fractions  having  the 
least  common  denominator. 

8.  When  is  it  necessary  to  reduce  fractions  having  different 
denominators  to  equivalent  fractions  having  a  common  de- 
nominator ? 

9.  A  horse  traveled  48^^  miles  in  one  day,  56^  the  next, 
40  J I  the  third,  and  45|J  the  fourth.  How  far  did  he  travel 
in  all  ? 

10.  From  a  bin  containing  25|  bushels  of  grain  there  were 
taken  out  5|  bushels  at  one  time  and  6|  bushels  at  another. 
How  much  remained  ? 

421.  1.  Find  the  greatest  common  divisor  of  36,  108,  and 
420. 

2.  Find  the  least  common  multiple  of  24,  180,  45,  and  60. 

3.  Required  the  amount  of  12j,  16|,  24f,  and  40f 

4.  Required  the  difference  between  84J  and  42j. 

5.  Multiply  f  of  12J  by  36f. 
6  Divide  27f  by  §  of  8 J. 

7.    Reduce  —  and  ^  to  simple  fractions. 

a    If  §  of  a  farm  is  worth  $  7000,  what  is  |  of  it  worth  ? 

9.  If  a  man  travels  240  miles  in  5 j  days,  how  far  will  he 
travel  in  3^  days  ? 

10.  A  coal-dealer  sold  f  of  what  coal  he  had  on  hand  for 
$  120,  at  the  rate  of  $  6  a  ton.     How  many  tons  had  he  ? 


278  EXAMINATION   QUESTIONS. 

422.  1.   Define  factoring.     Name  and  define  tlie  terms  of 
a  fraction. 

2.  When  are  two  or  more  numbers  said  to  be  prime  to  each 
other  ? 

3.  E,educe  the  following  by  cancellation  : 

800  X  378  X  44  X  15 
160  X   63   X  11  X  4 

4.  E-educe  f  to  a  fraction  having  135  for  its  denominator. 

5.  The  sum  of  two  numbers  is  36f-,  and  one  of  them  is  15f . 
What  is  the  other  ? 

6.  Divide  12  J  by  J  of  17. 

7.  What  is  the  value  of  an  estate  if  §  of  f  of  it  is  worth 
$45000? 

8.  What  is  the  value   of  |  of  a  farm  if  f  of  it  is  worth 
$4000? 

9.  Eeduce  j ^  to  a  simple  fr.iction. 

XO.    Is  ^4f\-{^  a  large  or  a  small  fraction,  and  why  ? 

423.  1.    Change  fgg-^,  j%^^,  4f|,  to  lowest  terms. 

2.  Add  4^y,  3|,  4-1,  f . 

3.  Take  12f  from  31|. 

4.  Having  $  4283}f,  I  gave  $  1597||  to  my  son.     How 
much  had  I  left  ? 

5.  Multiply  641  by  5|. 

6.  Keduce  ^  of  3}  of  4]  of  63§  of  |^  to  a  simple  fraction 

7.  —  plus  71  equals  what  ? 

8.  If  3^  lb.  cost  $  21,  what  will  3  lb.  cost  ? 

9.  5^j  is  contained  how  man}'  times  in  2^  ? 

10.    If  a  bird  can  fly  10}  miles  in  \  of  an  hour,  how  far  can 
it  fly  in  2  J  hours  ? 


EXAMINATION    QUESTIONS.  279 

424.  1.   Find  the  sum  and  the  difference  of   12|:J  and 

2.  In  what  two  ways  can  a  fraction  be  divided  by  a  whole 
number,  and  which  is  the  better  way  ? 

3.  If  a  man  can  travel  32 1  miles  in  one  day,  how  many  miles 
can  he  travel  in  5^  days  ? 

4.  A  man  received  $  84|  for  laboring  18|  days.  How  much 
did  he  receive  each  day  ? 

5  What  is  the  value  of  a  farm  if  f  of  it  is  worth  $  500  more 
than  f  of  it  ? 

6.  If  a  merchant  who  owns  f  of  a  store  should  sell  J  of  his 
share  for  $  12000,  what  is  the  value  of  the  whole  store  at  that 
rate? 

7.  A  has  $  12000.  If  $  600  be  added  to  f  of  A's  money,  the 
sum  will  equal  J  of  B's  mone3^     How  much  money  has  B  ? 

8.  Reduce  pr^ -^  to  a  simple  fraction. 

9.  A  man  who  owned  100  acres  of  land  sold  37i  acres  to 
one  man  and  ^  of  the  remainder  to  another.  How  many  acres 
had  he  left  ? 

10.  A  man  who  owned  |  of  f  of  a  ship  sold  J  of  his  inter- 
est for  $  32000.  What  was  the  value  of  the  whole  ship  at  the 
same  rate  ? 

425.  1.  Three  cheeses  weigh,  respectively,  46f,  49y^^,  and 
57|  lb.    What  is  their  entire  weight  ? 

2.  What  number  is  that  from  which  if  28-f-  is  taken  the  re- 
mainder is  65r| J  ? 


3.  4H  4- 56- 24^ -41H  =  ? 

4.  Find  the  cost  of  |  of  156§  acres  of  land  at  |  of  $54  an 
acre. 

5.  What  number  multiplied  by  33f  will  produce  297 J  ? 

6.  In  2i  acres  of  land,  how  many  building  lots  of  |  of  an 
acre  each  ? 


280  EXAMINATION   QUESTIONS. 

7.  Find  the  value  of  f   ^  ^t  - 

tV  X  5i 

8.  What  part  of  9  miles  is  f  of  8  miles  ? 

9.  §  of  a  farm  is  worth  $  9000.     Wliat  is  yV  ^^  i^  worth  ? 

10.  If  2  be  added  to  both  terms  of  the  fraction  J,  will  the 
value  be  increased  or  diminished,  and  how  much  ? 

426.  1.   (7i  -  3i)  -  145. 

2.  At  $  3  a  day  for  work,  what  part  of  a  day's  work  can  be 
had  for  $2.50? 

3.  Multiply  94  by  26§,  and  from  the  product  subtract  the 
quotient  of  12000  divided  by  7|. 

4.  Give  an  example  in  solving  which  it  is  necessary  to  use 
the  least  common  multiple  of  63  and  84. 

5.  The  denominator  of  a  certain  fraction  is  J  of  f  -f  J  X  3J^, 
and  the  numerator  is  |  of  the  denominator.  What  is  the  value 
of  the  fraction  ? 

6.  Find  the  amount  of  the  sum  and  the  difference  of  6§  X 
2f  and  5j  -^  3|. 

7.  A  boy  sold  a  book  for  $  2 J,  which  was  J  of  the  cost.  What 
did  he  lose  by  the  bargain  ? 

8.  Emma,  having  2  quarts  of  berries,  ate  J  of  them,  sold  J  of 
a  quart,  and  divided  the  remainder  equally  among  three  friends. 
What  did  each  friend  receive  ? 

9.  Multiply  24|  by  |.  Divide  24^  by  ^.  How  much  does 
the  quotient  exceed  tlie  product  ? 

10.  A  bought  f  of  a  farm  for  $1760.  B  bought  1  the  re- 
mainder at  the  same  rate,  and  C  took  the  remainder  at  an  ad- 
vance of  $  375.     What  did  C's  share  cost  ? 

DECIMALS. 

427.  1.   Change  ^\  to  an  equivalent  decimal. 

2.  From  eight  hundred  thousandths  take  eight  hundred 
thousandths. 

3.  Write  in  words  :  7.008  ;  9090.909 ;  0.00042. 


EXAMINATION    QUESTIONS.  281 

4.  Change  to  an  equivalent  common  fraction  in  its  smallest 
terms  0.0025. 

5.  Why  is  it  unnecessary  to  express  the  denominator  of  a 
decimal  f  I'action  ? 

6.  From  a  piece  of  cloth  containing  49  J  yd.  I  sold  ^  of  a  yd., 
3f  of  a  yd.,  and  21.125  yd.  What  was  the  length  of  the  rem- 
nant ? 

7.  Eeduce  |,  f,  ^,  and  ^^  to  decimals,  and  add. 

8.  How  much  will  240  men  earn  at  $  1.37  J  a  day  ? 

9.  If  4  bushels  of  beans  cost  $  12.56,  what  will  9  bushels 
cost? 

10.  What  part  of  the  quotient  of  24  thousandths  divided 
by  25  ten-thousandths  is  the  product  of  16  thousandths  by 
3  hundred  ? 

428.  1.    Change  ^J§^  to  a  decimal. 

2.  Change  960  hundred-thousandths  to  a  common  fraction 
in  smallest  terms. 

3.  Multiply  the  quotient  of  144  -^  12000  by  the  quotient  of 
.0144  -^  .00012. 

4.  Subtract  23  ten-millionths  from  .02  of  .006. 

5.  Divide  17.28  by  .0831 

6.  Multiply  .027  J  by  .36|. 

71 

7.  Change  ^^r^  to  a  decimal. 

8.  Tell  the  shortest  way  of  dividing  a  decimal  by  1000.  Of 
multiplying  it  by  100. 

9.  If  .03  of  a  number  is  90,  what  is  .005  of  it  ? 

10.   Divide  1.2  by  .0025,  and  subtract  the  divisor  from  the 
quotient. 

429.  1.  Change  -^^  to  a  decimal,  multiply  by  .0008,  and 
divide  the  product  by  .02. 

2.  Divide  one  thousand  by  one  thousandth,  and  from  the 
quotient  subtract  the  dividend,  the  divisor,  and  their  product. 


282  EXAMINATION    QUESTIONS. 

3.  Multiply  six  hundred  thousandths  by  six  hundred-thou- 
sandths, and  divide  the  product  by  .02j. 

4.  All  of  the  six  U.  S.  gold  coins  are  equal  in  value  to  how 
many  twentj^-five  cent  piece*? 

5.  Bought  15280  bricks  at  $  40  per  thousand.  350  were  so 
broken  that  they  were  worthless.  What  was  the  actual  cost 
per  thousand  of  those  used  ? 

6.  If  a  man  receives  $  1500  a  year  for  labor,  and  his  ex- 
penses  are  $  968,  in  what  time  can  he  save  enough  to  buy  a 
farm  worth  $  3724  ? 

7.  A  man  lost  0.60  of  his  money,  and  then  earned  $  130, 
when  he  had  0.83J  of  the  original  amount.  How  much  had 
he  at  first  ? 

8.  What  cost  19375  ft.  of  lumber  at  $  17.25  per  thousand  ? 

9.  Find  the  amount  of  the  following  purchases :  21|-  yd. 
carpeting  at  $1.75;  25  yd.  lining,  at  12^/:  2|  yd.  silk  at 
$  2.25,  and  f  yd.  velvet  at  $  2.87^. 

10.  If  my  gas  bill  was  1 11  when  I  burned  4400  feet  of  gas, 
what  will  it  be  when  gas  costs  I  more  and  I  burn  1500  feet 
less? 

COMPOUND    NUMBERS. 

430.     1.   Find  the  difference  between  25t\  and  15.064. 

2.  Define  a  complex  fraction ;  a  mixed  decimal ;  a  denom- 
inate number ;  reduction. 

3.  Change  ^ !}  to  a  complex  decimal  of  five  places,  and  0.0875 
to  a  common  fraction. 

4.  Eeduce  65  rd.  2  yd.  1  ft.  to  the  decimal  of  a  mile. 

5.  Reduce  0.5473  of  a  pound  troy  to  lower  denominations- 

6.  At  $  5.50  a  cord,  what  is  the  value  of  a  pile  of  wood  80 
feet  long,  12  feet  wide,  and  4  feet  high  ? 

7.  A  man  bought  a  farm  containing  260  acres  45  square 
rods  at  $  25.75  an  acre.     What  was  the  cost  of  the  farm? 

8.  In  16.45/>  metric  tons  how  many  kilograms  ? 


EXAMINATION   QUESTIONS.  283 

9.  A  man  paid  $  3.46  for  sugar  and  coffee ;  he  bought  6  lb. 
5  oz.  of  coffee  at  32  cents  a  pound,  and  paid  11  cents  a  pound 
for  the  sugar.     How  much  sugar  did  he  buy  ? 

10.  If  an  average  degree  of  latitude  is  69/oV  common  miles, 
and  a  meter  is  one  ten-millionth  of  the  distance  from  the 
equator  to  the  pole,  measured  on  a  meridian,  what  is  the 
length  of  a  meter  in  inches  ? 

431.  1.  How  many  years,  months,  and  days  from  the  dis- 
covery of  America,  Oct.  11,  1492,  to  the  Declaration  of  Inde- 
pendence, July  4,  1776  ? 

2.  A  farmer  had  two  farms,  one  of  104  A.  117  sq.  rd.,  the 
other  of  87  A.  78  sq.  rd.  He  reserved  40  A.  40  sq.  rd.,  and  di- 
vided the  remainder  equally  among  his  3  sons.  What  was 
the  share  of  each  son  ? 

3.  If  a  car  runs  16  mi.  25  rd.  12  ft.  in  40  minutes,  how  far 
will  it  run  at  the  same  rate  in  24  hours  ? 

4.  If  the  difference  in  the  time  of  Greenwich  and  of  St. 
Louis  is  5  h.  55  min.,  what  is  the  difference  of  longitude  ? 

5.  The  aggregate  weight  of  85  hogsheads  of  sugar  is  39  T. 
1625  lb.     What  is  the  average  weight  per  hogshead  ? 

6.  A  ship  sails  east  from  Boston,  longitude  71°  west,  2° 
30^  20'^  a  day.  What  is  her  longitude  at  the  end  of  5 
days  ? 

7.  Define  a  square  ;  a  rectangle  ;  a  cube  ;  a  solid. 

8.  If  a  man  wastes  4  minutes  a  day,  how  many  days,  hours, 
and  minutes  will  he  waste  in  the  years  1880  and  1881  ? 

9.  There  is  a  fence  enclosing  a  circular  field  32  feet  in 
diameter.  What  will  be  the  area  of  a  square  field  which  the 
same  fence  will  exactly  surround  ? 

10.  How  many  cubic  feet  of  water  must  be  drawn  from  a 
reservoir  24  ft.  6  in.  long  and  20  It.  9  in.  wide,  to  lower  the 
gurface  3  inches? 


284  EXAMINATION    QUESTIONS. 

432.  1.  A  room  measures  14  ft.  by  16  ft.,  and  is  8  ft.  high. 
How  many  rolls  of  paper  1 J  ft.  wide  will  cover  the  walls,  there 
being  8  yd.  in  a  roll  and  J  allowed  for  openings  ? 

2.    Divide  2|  times  |  of  29^  by  4|  times  ^j  of  8. 
a    Change  f  of  a  great  gross  to  integers  of  lower  denomina- 
tions. 

4.  How  many  square  inches  in  the  entire  surface  of  15 
bricks,  each  8  inches  lone,  4  inches  wide,  and  2  inches  thick  ? 

5.  What  date  comes  275  days  after  May  7  ? 

6.  What  is  the  cost  of  17  gal.  3  qt.  1  pt.  at  $  0.45  per  gallon. 

7.  3.75  X  48.341  -^.5^  =  ? 

8.  How  many  acres  in  ^  of  a  mile  of  street  60  ft.  wide  ? 

9.  A  room  10  ft.  high  contains  30000  cu.  ft.  What  will  it 
cost  to  carpet  it  at  $  .75  per  sq.  yd.  ? 

10.  How  many  square  yards  of  silk  in  300  yd.  of  3-inch 
ribbon  ? 

433.  1.  A  man  paid  $  660  for  a  piece  of  land  8  rods  long 
and  10  feet  wide,  and  sold  it  at  60  cents  per  square  foot. 
What  did  he  gain  ? 

2.  What  will  7bu.  3pk.  4qt.  of  nuts  cost  at  $4.80  per 
bushel  ? 

3.  It  takes  30  yards  of  carpeting  |  yd.  wide  to  carpet  a 
room  15  ft.  long.     How  wide  is  the  room  ? 

4.  Bought  an  acre  of  land  for  $  300,  sold  J  of  it  at  30  f  per 
square  foot,  and  the  remainder  at  cost.     What  did  I  gain  ? 

5.  Add  0.24  lb.  and  7  oz.,  avoirdupois. 

6.  What  will  12750  feet  of  boards  cost  at  $  27.50  per  thou- 
sand ? 

7.  What  part  of  an  acre  is  a  lot  of  land  132  ft.  long  and 
66  ft.  wide  ? 

8.  How  many  bags,  each  holding  2  bu.  1  pk.  3  qt.,  will  it 
take  to  hold  124  bu.  0pk.'7  qt.  ? 

9.  Find  the  exact  number  of  days  and  hours  from  9  o'clock 
K.  M.,  Jan.  7,  1876,  to  3  p.  m.  of  March  1,  1881. 


EXAMINATION    QUESTIONS.  285 

10.  How  many  building  lots,  each  75  ft.  by  125  ft.,  can  be 
made  out  of  1  A.  46  sq.  rd.  18|  sq.  yd.  of  land  ? 

434.  1.  Mr.  Howes  bought  a  farm,  198  rods  long  and  150 
rods  wide,  and  paid  $  32  an  acre.     What  did  it  cost  him  ? 

2.  I  bought  a  board  12  feet  long,  16  inches  wide  at  one  end 
and  9  at  the  other,  at  $  30  per  M.     What  did  it  cost  me  ? 

3.  How  many  cubic  feet  of  snow  will  there  be  on  an  acre 
of  land,  if  it  is  uniformly  6  inches  deep  ? 

4.  In  a  piece  of  ribbon  2 J  inches  wide  and  167  fset  long, 
how  many  square  yards  ? 

5.  Of  a  street  50  feet  wide,  how  many  feet  in  length  make 
one  acre? 

6.  How  many  bricks  (each  8  inches  long,  4  inches  wide, 
and  2  inches  thick)  will  make  a  cubical  pile  13  feet  each  way  ? 

7.  If  a  ship  sails  at  the  rate  of  2  miles  in  11  minutes  and 
11  seconds,  how  many  days  will  she  require  to  cross  the  At- 
lantic where  it  is  2979  miles  wide  ? 

8.  I  own  five  contiguous  unfenced  house-lots.  Each  lot  is 
50  ft.  wide  and  150  ft.  deep.  How  many  feet  of  boards  will 
enclose  said  lots  with  a  tight  board  fence  4  ft.  high,  and  what 
will  they  cost  at  $  16  per  M.  ? 

9.  A  grocer  uses  a  false  gallon  containing  3  qt.  IJ  pt.  What 
is  the  actual  worth  of  the  liquor  he  sells  for  $  240,  and  what 
does  he  make  by  the  cheat  ? 

10.  What  costs  a  pile  of  wood  17  ft.  8  in.  long,  8  ft.  wide, 
and  8  ft.  3  in.  high,  at  $  8.32  per  cord  ? 

435.  1.  What  part  of  an  acre  is  a  rectangular  piece  of 
land,  12  rods  long  and  110  feet  wide  ? 

2.  A  meter  is  39.37  inches.  How  many  meters  are  there 
in  a  mile  ? 

3.  Gold  is  19.35  times  as  heavy  as  water.  What  is  the 
weight  in  kilograms  of  a  cubic  meter  of  gold  ? 


286  EXAMINATION    QUESTIONS. 

4.  What  is  the  cost  of  25  yards  2  feet  3  inches  of  tubing  at 
$  0.25  per  yard  ? 

5.  Paid  $  5  for  constructing  2|  rods  of  stone  wall.  What 
will  a  wall  .875  of  a  mile  in  length  cost  at  the  same  rate  ? 

6.  Knowing  the  length  of  a  meter,  how  can  you  find  the 
length  of  a  degree  of  latitude  ?  Write  out  the  statement,  but 
do  not  perform  the  work. 

7.  A  pile  of  wood  8  feet  wide  and  4  feet  high  contains  forty 
cords.     How  long  is  it  ? 

8.  How  many  feet  of  boards  will  it  take  to  cover  the  walls 
of  a  house  56  feet  long,  25  feet  wide,  andv30  feet  high  ?  How 
much  will  they  cost  at  $  10  per  thousand  feet  ? 

9.  What  will  it  cost  to  make  a  sidewalk  8  ft.  wide  and 
225  ft.  long,  when  it  costs  $  30  to  make  one  10  ft.  wide  and 
90  ft.  long  ? 

10.  What  will  be  the  cost  of  constructing  a  railroad  25 
miles  145  rods  long,  at  $  700  per  mile  ? 

436.  1.  How  many  yards  of  carpeting  |  yd.  wide  will  carpet 
a  room  16^ feet  long  and  14  feet  9  inches  wide? 

2.  The  signal  service  reports  4^  inches  of  rain  as  falling  in 
24  hours.     How  many  cubic  yards  fell  on  J  of  an  acre  ? 

3.  Two  ships  on  the  ocean,  which  are  120  miles  apart,  are 
sailing  towards  each  other,  one  8|  miles  an  hour,  the  other 
lOf  miles  an  hour.  How  far  apart  will  they  be  in  4  hours 
and  40  minutes  ? 

4.  What  decimal  of  a  mile  is  575f  feet  ? 

5.  A  cubic  foot  of  water  weighs  62^  pounds  avoirdupois. 
The  specific  gravity  of  gold  is  19.25.  One  pound  avoirdupois 
equals  7000  grains  troy.  How  many  grains  does  a  cubic  inch 
of  gold  weigh  ? 

6.  A  farmer  sold  71200  pounds  of  hay  at  $  22  a  ton,  and 
purchased  19625  feet  of  boards  at  $  15  a  thousand.  How  much 
money  had  he  remaining  ? 

7.  If  a  car  runs  16  miles  25  rods  12  feet  in  40  minutes,  how 
far  will  it  run  at  the  s^i?^e  rate  in  4  hours  ? 


EXAMINATION    QUESTIONS.  287 

8.  The  aggregate  weight  of  67  hhd.  of  sugar  is  39  T.  16  cwt. 
35  lb.     What  is  the  average  weight  per  hhd.  ? 

9.  If  the  difference  in  the  time  of  Greenwich  and  of  Boston 
is  4  hours  44  minutes,  what  is  the  longitude  of  Boston  ? 

10.  A  room  is  18  ft.  8  in.  long  and  10  ft.  6  in.  wide.  One 
kind  of  carpeting  three-fourths  of  a  yard  wide  can  be  obtained 
for  $2  per  yard ;  another  kind,  a  yard  wide,  can  be  obtained 
for  $  1.75  per  yard.  Which  kind  is  the  more  expensive,  and 
how  much  will  it  cost  to  carpet  the  room  with  each  ? 

437.  1.  A  field  containing  24  acres  is  80  rods  long.  What 
is  its  width  ? 

2.  What  is  the  difference  between  25  rods  square  and  25 
square  rods  ? 

3.  How  many  tons  of  ice  can  be  packed  in  an  ice-house  50 
feet  long,  20  feet  wide,  and  12  feet  high,  a  cubic  foot  of  ice 
weighing  58.5  lb.  ? 

4.  What  is  the  difference  in  time  between  two  places  whose 
difference  in  longitude  is  4°  40'  ? 

5.  If  1  ounce  of  sugar  costs  1  cent,  what  will  be  the  cost  of 
5  T.  9  cwt.  75  lb.  ? 

6.  I  have  a  piece  of  land  42  rods  long  and  6  rods  wide.  I 
wish  to  make  7  square  lots  of  equal  size.  What  will  be  the 
cost  of  boundary  and  cross  fences  at  $  2.37 J  a  rod  ? 

7.  What  is  the  value  of  a  pile  of  wood  12  feet  long,  8  feet 
wide,  and  6  feet  high,  at  $  4.50  per  cord  ? 

8.  Mont  Cenis  Tunnel,  which  connects  the  railways  of 
France  and  Italy,  is  7  miles  190  rods  long,  and  the  Hoosac 
Tunnel  is  25000  feet  long.  The  latter  is  what  decimal  of  the 
former  ? 

9.  What  is  the  cost  of  4  loads  of  coal  weighing  2436,  2150, 
1735,  and  3462  lb.,  respectively,  at  $5.25  per  ton  ? 

10.  My  house  is  on  a  corner  lot,  100  ft.  on  one  street  and 
75  ft.  on  the  other.  The  sidewalk  is  8  ft.  wide.  How  many 
cubic  feet  of  snow  do  I  shovel  in  clearing  my  walk  after  a 
15-inch  storm  ? 


288  EXAMINATION   QUESTIONS. 

PERCENTAGE. 

438.  1.  Express  the  following  as  common  fractions  :  6 J  %, 
12i%,  8i%,  16f%/66f%. 

2.  What  per  cent  is  /^,  ii  f,  ^,  |,  f,  f^? 

3.  Define  percentage  ;  base  of  percentage  ;  rate  per  cent. 

4.  A  man  owns  60  %  of  a  ship  and  sells  75  %  of  his  share. 
What  part  of  the  whole  ship  does  he  sell  ? 

5.  What  is  the  difference  between  87  J%  of  $  5000,  and  .87^% 
of  the  same  sum  ? 

6.  The  difference  in  the  time  of  two  places  is  37  J  %  of  a  day. 
How  many  degrees  apart  are  the  meridians  of  those  places  ? 

7.  How  much  money  has  a  merchant  on  deposit  if  S3^7o  of 
one  third  of  it  is  $  700  more  than  25  %  of  one  fourth  of  it  ? 

8.  A  man  sold  a  house  for  $  5000,  which  was  25  %  more  than 
it  cost  him.  What  would  have  been  his  gain  per  cent  if  he 
had  sold  it  for  $  6000  ? 

9.  A  broker  received  $  12000  for  the  purchase  of  bank  stock. 
The  brokerage  was  J  %  on  the  purchase.  What  did  he  pay  for 
the  stock,  and  what  was  the  brokerage  ? 

10.  What  is  62^  %  of  a  sum  of  money  if  |  of  it  is  $  1200 
more  than  §  of  it  ? 

439.  1.  A  man  had  %  5420  in  bank.  He  drew  out  15  %  of 
it,  and  afterward  37^  %  of  the  remainder.  How  much  money 
had  he  then  in  bank  ? 

2.  A  man  sold  160  acres  of  land  for  %  4563.20,  which  was 
8  %  less  than  it  cost.     What  did  it  cost  an  acre  ? 

3.  A  broker  received  %  45337.50  to  invest  in  stocks  after  de- 
ducting a  commission  of  2\  %.  What  amount  did  he  invest, 
and  what  was  his  commission  ? 

4.  A  man  owns  a  boat-loa^l  of  corn  valued  at  %  1800,  and 
insures  87^%  of  its  value  at  1§  %.  What  premium  does  he 
pay? 

5.  \  of  75  per  cent  is  what  per  cent  of  J  of  90  per  cent  ? 


EXAMINATION   QUESTIONS.  289 

6.  A  man  owning  f  of  a  ship  sold  45%  of  his  share  for 
$  36000.  What  part  of  the  ship  did  he  still  own,  and  what 
was  its  value  at  the  same  rate  ? 

7.  What  per  cent  of  a  mile  is  124  rods  2  yards  2  J  feet  ? 

8.  A  merchant,  after  paying  40  %  of  his  debts,  found  that 
$  4800  would  pay  75  %  of  the  remainder.  What  was  his  whole 
indebtedness  ? 

9.  An  agent  received  $  3675  to  lay  out,  after  deducting  his 
commission  of  2J%.  What  was  the  amount  of  his  commis- 
sion ? 

10.  What  was  the  value  of  the  goods  purchased,  and  what 
was  the  remittance,  when  the  commission  at  3  J  %  amounted  to 
$  92.80  ? 

440.     1.   What  per  cent  of  |  is  |  ? 

2.  A  bankrupt's  assets  are  $  45000,  and  his  liabilities  $67500. 
What  per  cent  can  he  pay  ? 

3.  Bought  a  horse  for  $  200.  What  must  I  ask  for  him  in 
order  to  gain  10  per  cent  and  still  fall  10  per  cent  on  the  ask- 
ing price  ? 

4.  The  premium  for  insuring  a  cargo  at  2 J-  per  cent  was 
$  1000.    What  was  the  amount  insured  ? 

5.  If  f  of  a  barrel  of  flour  is  sold  for  what  f  of  the  barrel 
cost,  what  per  cent  is  gained  ? 

6.  A  merchant  paid  $  2500  for  cotton,  and  sold  it  at  10  per 
cent  advance,  taking  his  pay  in  prints,  which  he  sold  at  a  loss 
of  10  per  cent.     Did  he  gain  or  lose,  and  how  much  ? 

7.  Sold  2  horses  for  $  450  each,  thereby  gaining  25  per 
cent  on  the  one,  and  losing  25  per  cent  on  the  other.  What 
was  the  per  cent  of  gain  or  loss  on  the  investment  ? 

8.  What  per  cent  of  $  500  is  37^  %  of  $  1200  ? 

9.  A  merchant  reduced  the  price  of  a  piece  of  cloth  18  cents 
per  yard,  and  thereby  reduced  his  profit  on  the  cloth  from 
12^%  to  8  %.     What  was  the  cost  of  the  cloth  per  yard  ? 

10.  In  1864  the  greenback  dollar  was  worth  only  35f  cents 
in  gold.     What  was  the  price  of  gold  ? 

19 


290  EXAMINATION    QUESTIONS. 

441.  1.  A  owns  35  per  cent  of  a  steamboat  that  is  valued 
at  $  125000,  B  owns  45  per  cent  of  it,  and  C  owns  the  re- 
mainder.    What  is  the  value  of  each  man's  share  ? 

2.  A  farm  that  cost  %  4500  was  sold  for  %  5400.  What  was 
the  gain  per  cent  ? 

3.  A  merchant  sold  %  65000  worth  of  goods  in  a  year ;  40  per 
cent  of  the  receipts  were  sales  at  25  per  cent  profit,  and  the 
rest  at  30  per  cent  profit.     What  was  the  cost  of  the  goods  ? 

4.  An  agent  received  %  4500  with  which  to  purchase  mer- 
chandise, after  deducting  his  commission  at  2 J  per  cent.  How 
much  did  he  expend,  and  what  was  his  commission  ? 

5.  A  man  had  $  5000  in  bank.  He  drew  out  15  per  cent  of 
it,  then  20  per  cent  of  the  remainder,  and  afterward  deposited 
12J  per  cent  of  what  he  had  drawn.  How  much  had  he  then 
in  the  bank  ? 

6.  A  merchant  owes  %  15000,  and  his  assets  are  %  9525.  What 
can  he  pay  on  the  dollar  ? 

7.  If  by  selling  land  at  $  80  an  acre  I  lose  25  per  cent,  how 
must  I  sell  it  to  gain  40  per  cent  ? 

8.  What  must  be  paid  for  insuring  $4500  on  a  house,  at  | 
per  cent  ? 

9.  A  man  has  secured  a  policy  of  %  6000  on  his  life,  at  the 
rate  of  %  26.30  a  year  on  %  1000.  The  dividend  this  year  will 
reduce  his  payment  35  per  cent.  What,  therefore,  will  his 
payment  be  ? 

10.  A  grain  dealer  bought  corn  at  55  cents  a  bushel,  and 
sold  it  at  66  cents  ;  and  wheat  at  $1.10,  and  sold  it  at  %  1.37^. 
Upon  which  did  he  make  the  greater  profit,  and  how  much  ? 

442.  1.  How  many  feet  are  there  in  45  per  cent  of  a 
mile? 

2.  A  man  whose  income  was  %  2800  spent  %  1600.  What 
per  cent  of  his  income  remained  ? 

3.  A  man  invested  in  real  estate  %  7500,  which  was  37 J  per 
cent  of  his  property.     What  was  the  value  of  his  property  ? 


EXAMINATION    QUESTIONS.  291 

4.  A  has  $  1600.  75  per  cent  of  his  money  is  equal  to  62  J 
per  cent  of  B's  money.     How  much  have  both  together  ? 

5.  Alpheus  Cole  bought  12500  pounds  of  coal,  and  received 
a  discount  of  $  16.25,  or  20  %.     What  was  the  price  per  ton  ? 

6.  What  per  cent  of  1880  was  the  month  of  December  ? 

7.  A  merchant,  after  paying  40  %  of  his  debts,  found  that 
$  4800  would  pay  25  %  of  the  remainder.  What  was  his  whole 
indebtedness  ? 

8.  In  a  town  containing  2576  whites  77  %  of  the  inhabitants 
wefe  colored.     Eequired  the  population. 

9.  What  was  the  value  of  the  goods  purchased,  and  what, 
was  the  remittance,  wdien  the  commission  at  5^%  amounted 
to  1 92.80  ? 

10.  A  merchant  owning  45  %  of  a  ship  and  cargo  valued  at 
$  125000,  paid  4^%  for  insuring  his  share.  Eequired  the  base 
and  the  premium. 

443.  1.  Sold  tea  at  115  per  cent  of  its  cost,  and  thereby 
gained  9  cents  on  a  pound.     What  was  the  cost  per  pound  ? 

2.  What  sum  invested  at  4J-  per  cent  will  yield  an  annual 
income  of  $  900  ? 

3.  A  man  owns  a  house  worth  $  5000.  Eepairs  and  insur- 
ance average  1  per  cent  yearly.  Would  it  be  more  profitable 
for  him  to  rent  the  house  at  $  400  per  year,  or  to  sell  it  and 
invest  the  money  in  business  which  will  return  him  6  per  cent  ? 
How  much  ? 

4.  A  merchant  received  for  a  lot  of  goods  $  874.  He  had 
deducted  5  %  from  the  face  of  the  bill,  and  yet  found  he  had 
made  15%  on  his  investment.  What  did  he  pay  for  the 
goods  ? 

5.  A  merchant  sold  goods  for  $  5895,  and  gained  as  much  as 
he  would  have  lost  had  he  sold  them  for  14585.  What  was 
the  gain  per  cent  ? 

6.  The  premium  for  insuring  a  steamer  at  4J  per  cent  was 
$  2925.     What  was  the  value  of  §  of  the  steamer  ? 


292  EXAMINATION    QUESTIONS. 

7.  A  man  bought  a  lot  of  apples,  and  sold  them  for  20  per 
cent  more  than  they  cost,  by  which  he  gained  $24.80.  How 
much  did  they  cost  him  ? 

8.  By  selling  flour  at  $  6.65  per  barrel  I  shall  lose  5  per 
cent  of  its  cost.  For  how  much  must  I  sell  it  to  gain  5  per 
cent  ? 

9.  A  farmer  made  12  %  by  the  purchase  and  sale  of  a  horse> 
and  thus  gained  as  much  as  he  had  lost  by  selling  a  $  500  lot 
of  land  for  7J^%  less  than  its  value.  What  did  he  pay  and 
receive  for  the  horse  ? 

10.  By  selling  goods  at  $1537.90,  a  profit  of  12§  per  cent 
'  was  made.     What  per  cent  would  have  been  gained  or  lost  if 

they  had  sold  for  $  1651.65  ? 

INTEREST    AND    DISCOUNT. 

444.  1.  Find  the  amount  of  $  125  for  3  years  3  months 
3  days  at  7  %. 

2.  In  what  time,  at  8  %,  will  any  principal  double  itself  ? 

3.  What  principal,  at  7  %,  will  amount  to  $643.76  in  3 
years  4  months  24  days  ? 

4.  What  is  the  interest  of  $  475  from  January  1,  1880,  to 
July  4,  1882,  at  7^  per  cent  ? 

5.  What  is  the  difference  between  the  interest  and  true 
discount  of  $  450  for  1  year  4  months  at  6  per  cent  ? 

6.  The  difference  between  the  interest  of  $450  and  the 
interest  of  $  300  for  a  certain  time  is  $  15.30.  Rate,  6  per 
cent.     Required  the  time. 

7.  A  note  for  $  875,  dated  Jan.  1,  1879,  has  the  following 
indorsements :  March  10,  1880,  $  225 ;  April  1,  1881,  $  145. 
What  was  due  Dec.  31,  1881  ? 

8.  Required  the  avails  of  a  note  for  $  450,  due  in  6  months, 
discounted  at  a  bank  at  7^  per  cent. 

9.  The  proceeds  of  a  6  months'  note  discounted  at  a  bank 
at  6  per  cent  are  $  800.     Required  the  face  of  the  note. 


EXAMINATION    QUESTIONS.  293 

10.  How  much  more  is  the  interest  of  15  cents  for  15  years 
at  4  %  than  the  interest  of  15  dollars  for  15  days  at  8  %  ? 

445.  1.  In  what  time  will  the  interest  of  $  480  at  7  % 
equal  the  interest  of  $  356.50  for  3  years  9  months  25  days  at 
8%? 

2.  What  sum  of  money  will  gain  $  153.75  in  3  months  24 
days  at  7  %  ? 

3.  At  what  rate  per  cent  will  $  500  gain  $  84  in  2  years 
4  months  24  days  ? 

4.  What  is  the  accurate  interest  of  $1525  from  March 
20th  to  October  20th,  at  41  %  ? 

5.  What  is  the  compound  interest  of  $  1360  for  1  year  6 
months  at  8  %,  interest  compounding  semi-annually? 

6.  What  is  the  present  worth  and  the  true  discount  of 
$  1275,  due  in  1  year  5  months  18  days,  at  6  %  ? 

7.  What  is  a  negotiable  note  ?  How  may  a  note,  payable 
to  order,  be  made  negotiable  ? 

8.  When  is  a  note  due,  if  the  time  for  payment  is  not 
specified  ?  If  the  words  "  with  interest "  are  omitted,  when 
will  interest  accrue  ? 

9.  On  a  note  for  $  750  at  6  %,  dated  Jan.  15,  1878,  were 
the  following  indorsements :  Sept.  20,  1879,  $  250 ;  June  12, 
1880,  *  120.     What  was  due  May  25,  1881  ? 

10.  For  what  sum  must  a  note  be  drawn,  at  9  months  15 
days,  at  7  %,  so  that  the  proceeds,  when  discounted,  may  be 
1 1240  ? 

446.  1.  What  is  the  difference  between  the  simple  and 
the  compound  interest  on  $  700  for  2  years  6  months  at  7 
per  cent  ? 

2.  What  is  the  bank  discount  on  a  note  of  $  700  for  3 
months  at  the  rate  of  7|  per  cent  ? 

3.  What  principal,  on  interest  at  6  per  cent,  will  gain 
$  27.47  in  1  year  3  months  ? 


294  EXAMINATION   QUESTIONS. 

4.  At  what  rate  per  cent  must  $  648  be  on  interest,  to  gain 
$  81.873  in  2  years  3  months  17  days  ? 

5.  For  how  much  must  a  note,  payable  in  30  days,  be  given, 
that,  when  discounted  at  a  bank,  $  900  may  be  received  on  it, 
money  being  6  per  cent  ? 

6.  A  note  of  $  365  is  dated  July  1,  1878.  Indorsements ; 
Jan.  1,  1879,  $85;  July  1,  1879,  $125.  What  was  the 
amount  due  Jan.  1,  1881  ? 

7.  Find  the  compound  interest  of  $  245  for  2  years  6  months 
at  4^  %. 

8.  What  must  be  the  face  of  a  note  that,  when  discounted 
at  a  bank  for  5  months  at  8  %,  the  proceeds  may  be  $  217.35  ? 

9.  Find  the  bank  discount  and  proceeds  of  a  note  of  $450, 
payable  in  90  days,  discounted  at  8  %. 

10.  A  note  of  $  250,  dated  May  16,  1880,  and  payable  in 
4  months,  with  interest  at  6  %,  was  discounted  July  5,  1880, 
at  7  % .     What  were  the  proceeds  ? 

447.  1.  What  is  the  interest  of  $  105.23  at  6  per  cent, 
from  May  6,  1879,  to  July  7,  1881  ? 

2.  In  what  time  will  $  300  gain  $  47.25  at  6  per  cent? 

3.  The  interest  of  $  560  for  2  yr.  4  mo.  15  d.  was  $  106.40. 
What  was  the  rate  per  cent? 

4.  The  proceeds  of  a  note  discounted  at  a  bank  for  90  days, 
at  8  per  cent,  is  $293.80.     Kequired  the  face  of  the  note. 

5.  What  is  the  difference  between  the  simple  and  the  com- 
pound interest  on  $  650  for  2  yr.  8  mo.  at  6  per  cent  ? 

6.  What  principal  at  8  per  cent  will  gain  $78.08  in  4 
months  24  days  ? 

7.  If  the  interest  on  $  500  from  Jan.  6,  1880,  to  April  18, 

1880,  be  $12.75,  what  will  the  amount  of  that  principal  be, 
Feb.  23,  1881  ? 

8*  Note  given  for  $320,  July  14, 1874,  at  8  %  ;  payment,  Dec. 

24, 1874,  $  180  ;  settlement,  March  30,  1875.    What  was  due  ? 

9.   What  are  the  avails  of  a  note  for  $  1728,  due  Nov.  18, 

1881,  and  discounted  July  6,  1881  ? 


EXAMINATION   QUESTIONS.  295 

10.  $860  compounds  semi-annually  for  eighteen  months  at 
7  per  cent.     What  is  the  amount  ? 

448.  1.  At  6  %  how  long  will  it  take  $  175  to  amount  to 
$275? 

2.  A  note  of  $  500  on  interest  from  Jan.  10,  1881,  to  July 
10,  1881,  amounted  to  $  529.16|.     What  was  the  rate  ? 

3.  A  note  of  $  1250,  dated  July  5,  18G8,  was  paid  June  1, 
i870,  with  interest  at  8  per  cent.     What  was  paid  ? 

4.  What  is  the  difference  hetween  the  bank  discount  and  the 
true  discount  of  $  450  due  in  60  days,  discounted  at  6  per  cent  ? 

5.  On  a  note  for  $  425,  at  8  per  cent,  dated  March  25,  1880, 
were  the  following  indorsements  :  June  1, 1881,  $  75 ;  Dec.  30, 
1881,  $  120.     What  was  due  Sept.  1,  1882  ? 

6.  If  I  pay  a  debt  of  $  1410,  2  yr.  6  mo.  before  it  is  due, 
what  discount  should  be  made,  money  being  worth  7  %  ? 

7.  For  what  sum  must  a  note,  dated  May  10,  for  3  months, 
be  drawn,  to  yield  $  395.80,  if  discounted  at  a  bank  June  lOj 
money  being  worth  6  %  ? 

8.  June  15,  1880,  George  Page  borrowed  of  Henry  Smith 
$  2000j  and  gave  his  note  for  the  same,  with  interest,  at  8  per 
cent.  Aug.  27,  1881,  a  payment  of  $  1450  was  made,  and  a 
new  note  given  for  the  balance.  For  what  sum  was  the  new 
note  given  ?     Write  the  note  in  its  proper  form. 

9.  A  capitalist  holds  half  a  million  of  extended  U.  S. 
Government  5's,  now  paying  3 J  % .  What  is  his  quarterly 
income  ? 

10.  Sold  an  8  %  R.  E.  stock  which  cost  me  160,  and  bought 
Government  4's  at  120.  Did  I  increase  or  lessen  my  annual 
income  ? 

STOCKS  AND  AVERAGE  PAYMENTS. 

449.  1.  A  6  %  stock,  bought  at  120,  pays  what  per  cent  on 
the  investment  ? 

2.  What  sum  invested  at  4|  %  will  yield  an  annual  income 
of  $  1800  ? 


296  EXAMINATION    QUESTIONS. 

3.  Paid  $  9000  for  stock  at  10  %  below  par,  and  sold  it  at 
112.     What  per  cent  did  I  gain  ? 

4.  Bought  40  shares  of  railroad  stock,  par  value  $  100  per 
share,  at  33  %  below  par,  and  immediately  sold  them  at  20  % 
below  par.     What  per  cent  did  I  gain  by  the  transaction  ? 

5.  How  many  shares  of  B.  &  A.  E-.  R.  stock  can  be  bought 
for  $  25000  at  164|  and  brokerage  at  :J,  and  how  much  money 
will  be  left  ? 

6.  If  A  lends  B  $  300  for  4  months,  how  long  ought  B  to 
lend  A  $  800  to  equal  the  favor  ? 

7.  If  a  person,  owing  $  700,  payable  in  10  months,  pay 
$  300  down  and  $  200  at  the  end  of  6  months,  how  long  after 
the  end  of  10  months  may  he  delay  payment  of  the  balance  ? 

8.  A  merchant  owes  in  London  £  500  6  d.  How  much 
must  he  pay  for  a  draft  at  $4.86^  a  pound  sterling,  to  remit 
in  payment  ? 

9.  March  11,  1880,  a  merchant  sold  goods  to  the  amount  of 
$  1850,  on  a  credit  of  4  months.  April  7,  he  received  $  400 ; 
May  15,  $  270  ;  and  June  20,  $  350.  When  in  equity  should 
the  balance  be  paid  ? 

10.  Mr.  Adams  bought  goods  Aug.  1,  1881,  to  the  amount 
of  $  2400 ;  for  J  of  the  bill  he  was  to  pay  cash,  J  of  it  he  was 
to  pay  in  6  months,  and  the  balance  in  10  months.  On  what 
day  may  he  equitably  pay  the  whole  ? 

PROPORTION. 

460.  1.  The  ratio  is  2§,  the  antecedent  J  of  f .  What  is 
the  consequent  ? 

2.  At  the  rate  of  72  yards  for  £  44  16  s.,  how  many  yards 
of  cloth  can  be  bought  for  £  5  12  5.  ? 

3.  If  a  bin  8  ft.  long,  4^  ft.  wide,  and  2 J  ft.  deep,  holds  67i 
bu.,  how  wide,  must  another  bin  be  made,  that  is  18  ft.  long 
and  3g  ft.  deep  to  hold  450  bu.  ? 

4.  If  7  men  build  6^^  rods  of  wall  in  15|  days,  in  how  many 
days  can  12  men  do  as  much  ? 


EXAMINATION   QUESTIONS.  297 

5.  If  10  cents  will  buy  a  6-ounce  loaf,  when  flour  is  $11 
per  bbl.,  how  large  a  loaf  will  11  cents  buy,  when  flour  is 
$10perbbl.? 

6.  If  9  men,  by  working  8  hours  per  day,  can  mow  30  acres 
of  grass  in  2|  days,  how  many  acres  can  5  men  mow  in  3| 
days,  by  working  7|  hours  per  day  ? 

7a  A  vessel  has  provisions  for  50  men  for  4J  months,  allow 
ing  each  man  1|  lb.  per  day.  How  long  would  the  same  pro 
visions  furnish  75  men,  allowing  14  oz.  per  day  to  each  ? 

8.  If  36  men  can  dig  a  trench  60  rods  long  in  48  days, 
working  8  hours  a  day,  how  many  men  will  dig  a  trench  80 
rods  long  in  32  days,  working  6  hours  a  day  ? 

9.  If  19  men  build  a  wall  25  rods  long,  4  feet  thick,  and  3 
feet  high,  in  8  days,  working  8  hours  and  30  minutes  each 
day,  how  many  men  will  it  take  to  build  a  wall  45  rods  long, 
7|  feet  thick,  and  6  feet  high,  in  9  days,  working  9  hours  and 
30  minutes  each  day  ? 

10.  If  8  men  can  mow  36  acres  of  grass  in  9  days  of  9 
hours  each,  how  many  men  will  be  required  to  mow  48  acres 
in  12  days,  when  the  days  are  12  hours  long  ? 

451.  1.  A  and  B  are  partners  in  trade,  B  contributing  f 
of  the  capital.  What  is  A's  share  of  the  gain,  the  whole  gain 
being  $  4500  ? 

2.  Three  men  rent  a  pasture  for  $  55.  The  first  puts  in 
3  cows  2  months,  the  second  2  cows  4  months,  and  the  third  2 
horses  3  months.  Each  horse  eats  ^  more  than  a  cow.  What 
part  of  the  rent  should  each  man  pay  ? 

3.  Ames  and  Howe  entered  into  partnership  the  first  of 
January,  and  each  put  in  $  2060.  The  first  of  May,  Ames  put 
in  $  1000  more.  At  the  end  of  the  year  the  profits  proved  to 
be  1 2800.     What  should  each  receive  ? 

4.  Four  men  hired  a  pasture  containing  280  acres  at  $  1.25 
per  acre.  A  pastured  125  sheep ;  B,  150 ;  C,  200 ;  D,  225. 
How  much  rent  ought  each  to  pay  ? 


298  EXAMINATION    QUESTIONS. 

5.  A,  B,  and  C  gain  $  2250.  A's  gain  is  $  800,  and  B's 
$  1000.     C's  capital  is  $3000.     What  is  the  stock  of  each  ? 

6.  Jan.  1,  1880,  three  persons  began  business  with  $  1300 
furnished  by  A.  March  1,  B  put  in  $  1000 ;  August  1,  C  put 
in  $900.  The  profits  were,  at  the  end  of  the  year,  $750, 
Find  the  gain  of  each  partner. 

7.  Two  partners  engaged  in  business.  One  furnished  f  of 
the  whole  capital,  and  the  other  furnished  $  4000.  They  gained 
in  trade  20  per  cent  of  their  capital,  but  lost  $  500  from  bad 
debts.     What  was  each  partner's  share  of  the  net  gain  ? 

8.  Three  persons  engage  in  trade  with  a  joint  capital  of 
$  37680.  A  puts  in  $  6  as  often  as  B  puts  in  $  10,  and  as 
often  as  C  puts  in  $  14.  Their  annual  gain  is  equal  to  C's 
stock.     What  is  each  partner's  gain  ? 

9.  A  can  do  a  piece  of  work  in  6  days,  B  can  do  it  in  8  days, 
and  C  in  10  days.  In  what  time  can  they  do  it  by  working 
together  ? 

10.  A  man  owes  A  $  1800,  B  $  750,  and  C  $  1950.  His  as- 
sets are  :  money  on  hand,  $  205 ;  a  horse  and  carriage  valued 
at  $  260  ;  and  a  stock  of  goods  for  which  he  is  offered  $  1200. 
What  per  cent  can  he  pay,  and  what  will  A,  B,  and  C  each 
receive  ? 

ROOTS  AND  MENSURATION. 

452.     1.   Extract  the  square  root  of  925444. 


2.  va369  +  Va296  =  ? 

3.  If  a  line  160  feet  long  will  reach  from  the  top  of  a  steeple 
130  feet  high  to  the  opposite  side  of  the  street,  what  is  the 
width  of  the  street  ? 

4.  If  it  cost  $  312  to  enclose  a  field  216  rods  long  and  24 
rods  wide,  what  will  it  cost  to  enclose  a  square  field  of  equal 
area  with  the  same  kind  of  fence  ? 

5.  What  is  the  difference  between  the  cube  and  the  cube 
root  of  0.008  ? 


EXAMINATION   QUESTIONS.  299 

6.  In  the  center  of  a  square  garden  there  is  a  pond  covering 
an  area  of  810  square  feet,  which  is  a  tenth  of  the  whole  gar- 
den.    How  many  rods  of  fence  will  enclose  the  garden  ? 

7.  Extract  the  cube  root  of  34012224. 

8.  A  rectangular  piece  of  land  is  800  meters  long  and  600 
meters  wide.  If  you  walk  at  the  rate  of  a  hektometer  in  IJ 
minutes,  how  long  will  it  take  you  to  walk  from  one  corner  to 
another,  diagonally  across  the  piece  ? 

9.  What  are  the  dimensions  of  a  cube  that  has-  the  same 
volume  as  a  box  12  feet  6  inches  long,  10  feet  wide,  and  5  feet 
high?' 

10.  The  altitude  of  a  pyramid  having  a  square  base  is  80 
feet ;  the  length  of  each  side  of  the  base  is  120  feet.  Required 
its  slant  height,  surface,  and  contents. 

453.  1.  If  it  costs  $  75  to  paint  a  house  45  ft.  long,  what 
will  it  cost  to  paint  a  similar  house  60  ft.  long  ? 

2.  The  pedestal  of  a  certain  monument  is  a  cubical  block  of 
granite  containing  373248  cubic  inches.  What  is  the  length 
of  one  of  its  sides  ? 

3.  Two  flagstaffs,  one  80  feet  high  and  the  other  116  feet 
high,  stand  160  feet  apart  on  a  horizontal  plain.  What  is  the 
distance  between  their  tops  ? 

4.  In  a  square  field  of  one  acre,  if  a  man  mow  a  space  one 
rod  in  width  around  it  next  the  inner  border,  how  many  square 
rods  does  he  mow  ? 

5.  A  and  B  start  from  the  same  point,  each  walking  12 
hours  per  day.  A  walks  due  north,  at  the  rate  of  a  mile  in 
15  minutes,  while  B  walks  due  east,  at  the  rate  of  a  mile  in 
12  minutes.  How  far  apart  will  they  be  at  the  end  of  the 
fourth  day  ? 

6.  Find  the  entire  surface  and  the  diagonal  of  a  cube  con- 
taining 262144  cubic  inches. 

7.  How  many  globes  4  inches  in  diameter  will  it  take  to 
equal  in  volume  a  globe  12  inches  in  diameter  ? 


300  EXAMINATION   QUESTIONS. 

8.  How  much  more  ground  will  a  board  20  ft.  long,  2  ft. 
wide,  protect  from  rain  falling  vertically,  when  it  is  flat  on  the 
ground,  than  when  one  end  is  raised  12  ft.  ? 

9.  Two  rafters,  each  35  feet  long,  meet  at  the  ridgepole  of  a 
roof  15  feet  above  the  attic  floor.  What  is  the  width  of  the 
house  ? 

10.  I  have  a  square  garden,  100  feet  on  a  side  ;  I  wish  to 
enclose  the  garden  by  a  ditch  4  feet  wide.  How  deep  must 
the  ditch  be  dug  that  the  earth  thrown  out  may  raise  the  whole 
surface  of  the  garden  one  foot  ? 

MISCELLANEOUS. 
454.     1,  Eeduce  to  a  simple  fraction  ^      ^ — TqT' 

Z4:^  —  lof 

2.  What  decimal  of  J  of  a  ton  is  |  of  a  cwt.  ? 

3.  A  man  bought  a  house  for  $  2000,  and  sold  it  for  25  % 
more  than  he  paid  for  it  and  12 J  %  less  than  he  asked  for  it. 
What  did  he  ask  for  it  ? 

4.  On  a  note  for  %  400,  at  7  %,  there  was  paid  %  100  annually 
for  3  years.  How  much  remained  due  3  years  4  months  from 
the  date  of  the  note  ? 

5.  Required  the  bank  discount  and  the  proceeds  of  a  note 
for  %  1250,  due  in  90  days  at  7  %. 

6.  Eequired  the  time  of  day,  provided  the  time  past  noon 
equals  §  of  the  time  to  midnight. 

7.  Find  the  cost  of  raising  the  surface  of  \  of  an  acre  9  inches, 
at  50  cents  per  cubic  yard  of  earth. 

a  It  costs  $202.80  to  enclose  a  field  108  rods  long  and  48 
rods  wide,  what  will  it  cost  to  enclose  a  square  field  of  equal 
area  with  the  same  kind  of  fence  ? 

9.    What  is  the  cube  root  of  74088  ? 
10.    How  many  days  will  21  men  require  to  dig  a  ditch  80 
feet  long,  8  feet  wide,  and  4  feet  deep,  if  7  men  can  dig  a 
ditch  60  feet  long,  6  feet  wide,  and  3  feet  deep  in  12  days  ? 


EXAMINATION   QUESTIONS.  301 

455.  1.  How  much  cloth  J  of  a  yard  wide  will  cover  27 
tables  6  ft.  long  and  3  ft.  2  in.  wide  ? 

2.  What  is  the  amount  at  compound  interest  of  $  200,  at 
8  per  cent,  for  2  y.  6  mo.  6  d.  ? 

3.  What  is  the  length  of  one  side  of  a  square  farm  contain- 
ing 102  A.  64  sq.  rd.  ? 

4.  What  is  the  length  of  a  cubical  block  of  granite  contain- 
ing 47|f  I  cubic  feet  ? 

5.  Divide  600  by  .012,  multiply  the  .quotient  by  .05,  and 
divide  .005  by  that  product. 

6.  The  first  term  of  a  proportion  is  (4f  -^  .03),  the  second 
is  6 1,  and  the  fourth  is  8J-.     E/Cquired  the  third  term. 

7.  Bought  a  hogshead  of  wine  for  $  420  ;  but  10^  gallons 
having  leaked  out,  how  must  I  sell  the  remainder  per  gallon 
to  gain  25  per  cent  ? 

8.  What  is  the  interest  of  $  1200,  at  4J  per  cent,  from 
Aug.  18,  1880,  to  April  30,  1882  ? 

9.  If  25  men,  in  6  days  of  10  hours  each,  build  200  rods  of 
wall,  how  many  rods  will  be  built  by  12  men  in  5  days  of  8 
hours  each  ? 

10.  My  friend  lends  me  $  7000  for  15  days.  How  long 
must  I  lend  him  $  7500  to  requite  the  favor  ? 

456.  1.  In  the  number  78.342,  the  value  of  the  7  is  how 
many  times  as  great  as  the  value  of  the  2  ? 

31 
.2.  From -^  of  12i  take  4f. 

3.  A  merchant  who  owns  .36  of  a  ship  sells  85%  of  his 
share  for  $  22950.  What  is  the  value  of  the  whole  ship  at 
that  rate  ? 

4.  The  capital  stock  of  a  company  is  $  80000  ;  its  gross  earn- 
ings annually  are  1 8000,  and  its  expenses  $  3200.  What  per 
cent  on  his  investment  does  a  stockholder  receive  who  pur- 
chased at  20  %  above  par  ? 

5.  How  much  more  is  the  compound  than  the  simple  inter- 
est on  $  625  at  8  %  for  2  years  6  months  ? 


302  EXAMINATION   QUESTIONS. 

6.  Find  the  date  when  due,  the  bank  discount,  and  the 
proceeds  of  a  note  of  $  1250,  dated  Feb.  12,  1881,  payable  in 
90  days. 

7.  If  6  men  can  reap  14  acres  of  wheat  in  2  days  of  10  hours 
each,  in  how  many  days  of  8  hours  each  can  5  men  reap  25 
acres  ? 

8.  The  length  of  a  rectangular  field  containing  an  acre  is 
twice  its  width.     What  is  the  length  of  its  diagonal  ? 

9.  Find  the  solid .  contents  of  a  cone  whose  height  and 
diameter  are  each  12  inches. 

10.  The  length  of  a  meter  being  known,  how  can  the  length 
of  the  earth's  diameter  be  ascertained  ? 

457.  1.  I  hire  $  200  for  1  y.  8  mo.  18  d.  at  8  %,  at  simple 
interest,  and  loan  it  at  compound  interest.     Required  my  gain. 

2.  If  8  men  in  4  days  of  12  hours  each  can  mow  18  acres 
of  grass,  how  many  acres  can  6  men  mow  in  10  days  of  9 
hours  each  ? 

3.  Bought  a  house  for  $  10000  gold,  and  sold  it  for  $  15000 
currency.  If  gold  at  time  of  selling  is  worth  125,  what  is  my 
gain  per  cent  ? 

4.  How  many  gross  of  tacks  can  be  bought  for  $  12,  if  each 
tack  cost  .02  of  a  mill  ? 

5.  What  will  be  the  cost  of  a  close  fence,  5  J  ft.  high,  around 
a  field  14  rd.  long,  12 J  rd.  wide,  the  boards  costing  $42  per 
thousand  ? 

6.  Bought  a  piece  of  cloth  containing  72  yd.  for  $  280  less 
10  %,  and  sold  it  at  $  4.50  per  yd.,  receiving  in  payment  a  note 
on  90  days,  which  I  immediately  have  discounted  at  a  bank  at 
6  %.     Do  I  gain  or  lose,  and  how  much  ? 

7.  If  a  bullet  J  inch  in  diameter  weighs  IJ  oz.,  what  will 
be  the  weight  of  a  cannon-ball  7  inches  in  diameter  ? 

8.  What  is  the  area  of  one  side  of  a  cubical  block  contain- 
ing the  same  number  of  cubic  feet  as  one  tliat  is  27  ft.  by  8  ft. 
by  125  ft.  ? 


EXAMINATION   QUESTIONS.  303 

9.  Sold  20  loads  of  hay,  each  weighing  16  cwt.  40  lb.,  at 
$  35  per  ton.  How  long  must  what  I  receive  be  on  interest, 
at  8  %  simple  interest,  to  amount  to  as  much  as  would  have 
been  received  for  20  tons  ? 

10.  I  paid  $  4.50  for  a  line  that  would  just  reach  from  the 
top  of  a  spire  to  the  ground,  at  a  distance  of  56  ft.  from  the 
foot  of  the  spire ;  price  being  $  0.50  per  lb.,  allowing  18  ft.  to  a 
pound,  what  was  the  height  of  the  spire  ? 

458.  1.  After  ^  per  cent  of  a  flock  of  sheep  had  been  killed 
by  dogs,  and  68  had  been  sold  to  a  butcher,  ^  of  the  original 
flock  were  left.     Required  the  number  of  sheep  in  the  flock. 

2.  A  pile  of  4-foot  wood  256  ft.  long  and  5  ft.  high  was  sold 
for  $  152M,     What  did  a  cord  cost  ? 

3.  The  floor  of  a  public  house,  56  by  85  feet,  is  of  boards 
14  ft.  long,  6  in.  wide.  There  are  8  nails  in  each  board. 
Allowing  68  nails  to  a  pound,  how  many  pounds  are  in  the 
floor? 

4.  Divide  $  1000  between  two  children  aged,  respectively, 
6  years  and  8  years,  in  proportion  to  the  square  of  their  ages. 

5.  If  you  hire  $1200  for  1  y.  8  mo.  19  d.,  at  8  per  cent 
simple  interest,  and  lend  it  for  the  same  rate  and  time  at 
compound  interest,  payable  semi-annually,  how  much  do  you 
gain  ? 

6.  The  square  root  of  a  number  is  2304.  What  is  its  cube 
root? 

7.  A  |-in.  faucet  fills  a  cistern  in  3  hours.  How  long  will 
a  Ij-in.  faucet  require  ? 

8.  Exchange  is  2  per  cent  below  par.  How  large  a  60-day 
draft  can  you  buy  for  $  1939,  the  rate  of  discount  being  6  per 
cent? 

9.  At  what  rate  will  $  508.50  earn  $  89.609  in  2  y.  2  mo. 
13  d.? 

10.  Sent  my  agent  $  7315  with  which  to  buy  apples,  after 
deducting  4J  per  cent  commissioUc  What  per  cent  is  his  com- 
mission of  $  6300  ? 


304  EXAMINATION   QUESTIONS. 

459.  1.  What  would  be  my  annual  income  on  $  10000,  in- 
vested in  5  per  cent  bonds  at  5  per  cent  premium  ? 

2.  What  would  be  the  cost  of  a  section  of  U.  S.  land  con- 
taining 640  acres  at  $  1.50  per  centare  ? 

3.  Calling  the  earth's  diameter  7912.5  miles,  what  is  the 
'ength  in  miles  of  a  minute  on  the  equator  ? 

4.  Give  two  ways  of  changing  the  form  of  a  fraction  witb' 
out  changing  the  value,  and  explain  why  the  value  is  not 
changed. 

5.  What  will  it  cost  to  carpet  a  room  18  by  15  ft.  with  car-^ 
peting  27  in.  wide  at  $  2.50  per  yard  ? 

6.  Two  men  starting  at  different  points  travel  till  they  meet ; 
one  finds  his  watch  70  minutes  fast,  the  other  120  minutes  slow. 
How  far,  and  in  what  direction,  does  each  go  ? 

7.  Sold  two  farms  for  $  3000  each  ;  on  one  I  make  16 1  per 
cent,  and  on  the  other  I  lose  12  per  cent.  Find  the  net  gain 
or  loss. 

8.  How  much  must  be  invested  in  U.  S.  4's  at  120  to  pro- 
duce a  semi-annual  income  of  $  960  ? 

9.  A  man  having  a  field  40  rods  square  sold  to  A  100  square 
rods,  to  B  4  acres,  and  to  C  20  rods  square.  How  much  re- 
mained unsold  ? 

10.  What  per  cent  of  the  square  root  of  21316  is  the  cube 
root  of  150568768  ? 

460.  1.  What  will  be  the  cost  of  fencing  a  section  of  land 
at  12  4^  cents  per  foot  ? 

2.  If  40  yards  of  tapestry  will  carpet  a  room  18  feet  long,- 
what  is  its  width,  tapestry  being  |  yd.  wide  ? 

3.  Reduce  to  a  simple  fraction     .\    —         . 

18i  — 12f 

4.  What  is  the  length  of  a  pile  of  wood  containing  12  cords 
if  it^  width  is  9  feet  and  its  height  6  feet  ? 


EXAMINATION   QUESTIONS.  305 

5.  A  merchant  bought  45  pieces  of  cloth,  each  piece  con- 
taining 30  yards,  at  $  3.75  per  yard,  on  9  months'  credit,  and 
sold  the  same  immediately  at  $  4  per  yard,  on  4  months'  credit. 
What  was  his  gain  at  date  of  purchase  ? 

6.  Kequired  the  proceeds  of  a  note  for  $  4500,  dated  June 
15,  running  4  months,  and  discounted  August  18  at  7J%. 

7.  What  sum  of  money  will  amount  to  $520  in  3  years 
9  months,  at  4  %  ? 

8.  If  6  men  in  16  days  of  12  hours  each  build  a  wall  15  feet 
long,  8  feet  high,  and  3  feet  thick,  how  many  men  will  be  re- 
quired to  build  a  wall  45  feet  long,  9  feet  high,  and  6  feet  thick, 
in  24  days  of  9  hours  each  ? 

9.  What  is  the  distance,  in  rods,  from  the  center  to  each 
corner  of  a  section  of  land  ? 

10.  How  many  cubic  feet  of  water  must  be  drawn  from  a 
reservoir  24  feet  6  inches  long,  and  20  feet  9  inches  wide,  to 
lower  the  surface  8  inches  ? 

461.  1.  Write  a  note  due  in  90  days,  dated  March  30, 
1881,  and  signed  by  John  Smith,  on  which  Wm.  White  can 
obtain  $  850  at  a  bank,  the  bank  discounting  at  8  per  cent. 

2.  Find  the  difference  between  3J  divided  by  .31,  and  .31 
divided  by  3^. 

3.  For  what  must  hay  be  sold  per  ton  to  gain  18|:  per  cent, 
if  by  selling  at  $  46  per  ton  25  per  cent  be  gained  ? 

4.  At  $  50  an  acre,  what  will  be  the  value  of  a  field  which 
can  be  divided  into  12  lots,  each  80  feet  square  ? 

5.  How  many  reams  of  paper  will  be  required  to  make  32 
octavo  books,  each  containing  348  pages  ? 

6.  A  commission  merchant  receives  $  720  to  be  expended  in 
butter,  reserving  his  commission  of  2J  per  cent  on  the  amount 
expended.  He  pays  37  cents  per  pound.  How  many  pounds 
can  he  buy  ? 

7.  A  sells  cloth  at  a  profit  of  9  per  cent ;  B  sells  at  a  profit 
of  7  per  cent,  but  sells  5  yards  while  A  is  selling  3.  Which 
will  make  a  greater  profit  on  a  capital  of  $  1000  ?  Give  the 
Teason  for  your  answer. 


306  EXAMINATION    QUESTIONS. 

8.  There  is  a  fence  enclosing  a  circular  field  48  feet  in  diam- 
eter. What  will  be  tlie  area  of  a  square  field  which  the  same 
fence  will  exactly  surround  ? 

9.  A  loans  $  325.50  for  3  y.  4  mo.  24  d.,  and  recei\'es  for  in- 
terest $  77.469.  B  loans  $  1.80,  and  his  money  amounts  in 
the  same  time  to  $2,259.  Which  receives  the  greater  per 
cent? 

10.  Find  the  difference  between  the  square  root  and  the  cube 
root  of  .0064 

462.  1.  What  will  37^  bushels  of  salt  cost,  if  13f  bushels 
cost  $  11.75  ? 

2.  How  many  stones  10  inches  long,  9  inches  wide,  and  4 
inches  thick,  will  it  take  to  build  a  wall  80  feet  long,  20  feet 
high,  and  2^  feet  thick  ? 

3.  If  you  buy  eggs  for  12|^  cents,  and  sell  them  for  15  cents 
per  dozen,  for  how  much  must  wood  be  sold,  that  cost  you  $4.20 
per  cord,  to  gain  in  the  same  ratio  ? 

4.  What  must  I  pay  for  25  shares  of  bank  stock,  at  7f  per 
cent  advance,  the  par  value  being  $  250  per  share  ? 

5.  For  how  much  less  than  its  face  should  a  note  of  $  840 
be  sold,  if  payable  in  4  months  ? 

6.  What  is  the  value  of  a  lot  of  land  in  the  form  of  a  right- 
angled  triangle,  the  base  being  15  yards  and  the  hypothenuse 
25  yards,  at  8J  cents  per  square  foot  ? 

7.  In  a  school-room  35  feet  long,  30  feet  wide,  and  12  feet 
high,  there  are  40  pupils,  each  breathing  10  cubic  feet  of  air 
per  minute.  In  what  time  will  the  air  be  unfit  for  respii  ation, 
if  no  pure  air  be  admitted  ? 

8.  A  man  bought  3  loads  of  hay.  The  first  load  weighed 
4832  lb.,  tare  1124  lb. ;  the  second  weighed  4628  lb.,  tare  1136 
lb. ;  the  third  weighed  4976  lb.,  tare  1142  lb.  What  did  it  cost 
him  at  $  33.37  i  per  ton  ? 

9  A  cargo  of  4000  bushels  of  wheat,  worth  $  1.20  per  bushel, 
is  insured  at  -J  of  1 J  per  cent  on  §  of  its  value.  If  the  cargo  be 
lost,  how  much  will  the  owner  of  the  wheat  lose  ? 


EXAMINATION   QUESTIONS.  307 

10.  Mr.  B.  mortgaged  his  farm  for  $  6000,  Oct.  1,  1879,  to 
be  paid  in  6  years,  with  interest  at  8  per  cent.  Three  months 
from  date  he  paid  I  500 ;  Sept.  10,  1880,  $  1126 ;  March  31, 
1881,  $  2000;  and  Aug.  10,  1881,  $876.50.  How  much  was 
due  at  the  expiration  of  the  time  ? 

463.  1.  What  cost  a  piece  of  land  70  rods  5|  feet  long,  52 
/  rods  8 J  feet  wide,  at  $  256  per  acre  ? 

2.  What  is  the  interest  of  $  725  for  9  months  4  days,  at  5 
per  cent  ? 

3.  Bought  a  farm  for  $  4200,  agreeing  to  pay  $  600  down 
and  the  rest  in  six  equal  semi-annual  installments.  When 
could  I  justly  make  one  payment  for  the  whole  ? 

4.  A  chimney  5  ft.  square  and  50  ft.  high  has  two  flues,  each 
1  ft.  square.  How  many  bricks,  8  in.  by  4  in.  by  2  in.,  were 
used  in  its  construction  ? 

5.  Having  used  my  carriage  three  years,  I  am  willing  to 
sell  it  at  a  loss  of  20  per  cent.  If  I  receive  $  250  for  it,  what 
was  the  cost  ? 

6.  What  is  the  circumference  of  the  largest  wheel  that  can 
be  got  through  a  door  5  ft.  wide  and  12  ft.  high  ? 

7.  A,  B,  C,  and  D  hired  a  pasture  for  $  75.  A  put  in  7 
cows,  6  weeks  ;  B  3  cows,  13  weeks ;  C  4  cows  and  9  sheep,  7 
weeks  ;  D  42  sheep,  5  weeks.  A  cow  would  eat  as  much  as 
3  sheep.     What  should  each  pay  ? 

8.  Divide  $  3000  among  A,  B,  and  C,  giving  A  $  50  more 
than  B,  and  B  $  250  more  than  C. 

9.  If  a  cistern  6  ft.  in  diameter  hold  80  barrels  of  water, 
what  must  be  the  diameter  of  a  cistern  of  the  same  depth  to 
hold  1280  barrels  ? 

10.  A  can  do  a  piece  of  work  in  |  of  an  hour ;  B  can  do  | 
of  it  in  one  hour.     In  what  time  can  both  do  it  ? 


308  APPENDIX. 


APPENDIX. 


ROMAN   NOTATION. 

464.    Roman  Notation  uses  seven   capital   letters  in  ex- 
pressing numbers : 

I,        V,        X,         L,  C,  D,  M. 

1,        5,        10,        50,       100,        500,        1000. 

All  other  numbers  are  expressed  by  repeating  or  combining 
these  letters,  according  to  the  following 

465.     Principles. 

1.  Repeating  I,  X,  C,  or  M,  repeats  its  value. 

Thus,    I    stands  for  one ;    II,   two ;    III,    three ;    X,    ten ; 
XX,  twenty  ;  XXX,  thirty,  etc. 

2.  When  I  is  placed  before  V  or  X,  X  before  L  or  C,  or  C 
before  M,  the  difference  of  their  values  is  expressed. 

Thus,  IV  stands   for   four ;    IX,   nine ;    XL,    forty ;    XC, 
ninety. 

3.  When  a  letter  is  placed  after  another  of  greater  vc^lue^ 
the  sum  of  their  values  is  expressed. 

Thus,  VI  stands  for  six;  XI,  eleven;  XV,  fifteen;  XXV 
twenty-five,  etc. 


APPENDIX. 


309 


4.    A  dash  ( — )  over  V,  X,  L,  C,  D,  or  M,  increases  the  value 
of  the  letter  a  thousand  fold. 

Thus,  V  stands  for  five  thousand ;  X,  ten  thousand ;  L,  fifty 
thousand. 

466.      TABLE. 


I 

1 

XIV 

14 

xo 

90 

II 

2 

XV 

15 

c 

100 

III 

3 

XVI 

16 

cc 

200 

IV 

4 

XVII 

17 

ccc 

300 

V 

5 

XVIII 

18 

CD 

400 

VI 

6 

XIX 

19 

D 

500 

VII 

7 

XX 

20 

DC 

600 

VIII 

8 

XXX 

30 

M 

1000 

IX 

9 

XL 

40 

MD 

1500 

X 

10 

L 

50 

C 

100000 

XI 

11 

LX 

60 

M 

1000000 

XII 

12 

LXX 

70 

MDCXX 

1620 

XIII 

13 

LXXX 

80 

MDCCCLXXXI 

1881 

Express 

infi 

gures  : 

1. 

XX] 

ax. 

4.   XMCCXXII. 

2. 

LX} 

CXIII. 

5.   MMDXLIV. 

3. 

CD] 

ax. 

6.  MDCCCXCVIII. 

Express 

in  h 

itters : 

7. 

Twenty-i 

line. 

10. 

One  hundred  sixty-one. 

8. 

Seventy- 

three. 

11. 

Fifteen  hundred  eighty. 

9. 

Ninety-e 

ight. 

12. 

Two  thousand  eight  hundred. 

13.   One  thousand  seven  hundred  seventy-six. 


FUNDAMENTAL.    PROCESSES, 

467.  The  Fundamental  Processes  of  arithmetic,  or  those 
upon  which  all  others  depend,  are  addition,  subtraction,  mul- 
tiplication, and  division. 

These  are  also  sometimes  called  the  ground  rules  of  arith- 
metic. 


310 


APPENDIX. 


PRIME    NUMBERS. 

468.    Ko  direct  method  of  detecting  prime  numbers  has 
been  discovered. 

The  prime  numbers  to  1009  are  included  in  the  following 

TABLE. 


1 

59 

139 

233 

337 

439 

657 

653 

769 

883 

2 

61 

149 

239 

347 

443 

663 

669 

773 

887 

3 

67 

151 

241 

349 

449 

569 

661 

787 

907 

6 

71 

157 

251 

363 

457 

571 

673 

797 

911 

7 

73 

163 

257 

359 

461 

677 

677 

809 

919 

11 

79 

167 

263 

367 

463 

587 

683 

811 

929 

13 

83 

173 

269 

373 

467 

693 

691 

821 

937 

17 

89 

179 

271 

379 

479 

599 

701 

823 

941 

19 

97 

181 

277 

383 

487 

601 

709 

827 

947 

23 

101 

191 

281 

389 

491 

607 

719 

829 

953 

29 

103 

198 

283 

397 

499 

613 

727 

839 

967 

31 

107 

197 

293 

401 

603 

617 

733 

863 

971 

37 

109 

199 

307 

409 

609 

619 

739 

857 

977 

41 

113 

211 

311 

419 

521 

631 

743 

859 

983 

43 

127 

223 

313 

421 

623 

641 

761 

863 

991 

47 

131 

227 

317 

431 

641 

643 

757 

877 

997 

53 

137 

229 

331 

433 

547 

647 

761 

881 

1009 

CIRCULATING-    DECIMALS. 

469.  A  Circulating  Decimal  is  a  decimal  in  which  a  figure 
or  a  set  of  figures  continually  repeats.     Thus, 

§  =  0.666  ...    and  T^j.  =  0.727272  . . . 

470.  A  Repetend  is  the  part  of  the  decimal  that  continu- 
ally repeats. 

It  may  be  marked,  when  a  single  figure  repeats,  by  a  dot 
(.)  over  it,  and  when  a  set  of  figures,  by  a  dot  over  the  first 
and  the  last  of  the  set.     Thus, 


APPENDIX.  311 

0.666 . . .  =  0.6,  read  repetend  6  j  and  0.727272  ...  =  0.72, 
read  repetend  72. 

471.  A  Pure  Circulate  has  in  the  decimal  no  figure  but 
the  repetend.     Thus, 

0.93  is  a  pure  circulate. 

472.  A  Mixed  Circulate  has  in  the"  decimal  a  part  before 
the  circulate.     Thus, 

0.936  is  a  mixed  circulate. 

473.  A  Circulating  Decimal  arises  from  the  reduction  to  a 
decimal  of  a  common  fraction  whose  denominator  contains 
other  prime  factors  than  2  or  5. 

14.  Express  the  circulate  0.93  as  a  common  fraction  in  its 
lowest  terms. 

0.93  X  100  =  93.93 .  . .  Solution.  — As  the  repetend  has 

0.93  X       1  =    0.93  .  .  .  ^wo  places  of  figures,  we  multiply 

JTaXT;     ^  _  ^  it  by  100,  and  have  93.93.     Sub- 

tracting from  this  once  0.93  gives 
0.93  =  _-  =  — .  ^  number  with  the  same  figures  as 

99       oo  the  circulate,  but  which  do  not  re- 

peat.   Thus,  99  times  the  circulate  =  93,  and  once  the  circulate  must 
beff,  orfj.     That  is, 

474.  A  repetend  is  equal  to  a  common  fraction  whose  nu- 
merator is  the  figures  of  the  repetend,  and  the  denominator 
as  many  9'5  as  there  are  figures  in  the  repetend. 

15.  Express  the  circulate  0.093  as  a  common  fraction  in  its 
simple  form. 

Solution.  —  0.093  =  0.09|-  =.^  =  —  , 
^        100       75 

Beduce  to  common  fractions  in  their  simplest  form : 

16.  0.753.  18.  0.425.  20.   0.135. 

17.  0.594.  19.  7.345.  21.   53,00243. 


312  APPENDIX. 

22.  What  decimal  will  express  the  difference  between  29.259 

and  25.047  ? 

23.  What  decimal  will  express  the  product  of  5.9  by  0.08  ? 

24.  What  decimal  will  express  the  quotient  of  4.27  divided 
by  0.42  ? 

.  TABLES. 

475.    Surveyors'  Measures. 

7.92  inches  are  1  link,  1. 
25  links      "    1  rod,  rd. 
4  rods       "    1  chain,  ch. 
80  chains   "    1  mile,  mi. 


10  square  chains  are  1  acre. 

Note.  —  A  Gunter's  Chain  is  the  unit  of  measure,  and  is  4  rods,  or  66  feet 
long,  and  consists  of  100  links. 

476.    Mariners'  Measures. 

6    feet  are  1  fathom. 

120    fathoms  "   1  cahle-length. 

1\  cable-lengths    "   1  mile. 


1.15  common  miles  are  1  nautical  mile. 
3  nautical  miles    "   1  marine  league. 

477.    Apothecaries'  Measures. 

Weight.  Liquid. 


20  grains    are  1  scruple,  9. 
3  scruples  "   1  dram,    3 
8  drams      "    1  ounce,  S  . 

12  ounces     "   1  pound,  ft. 


60  minims  ("it\^)  are  1  fluid  drachm,  f.  5 
8  fluid  drachms  '*    1  fluid  ounce,  f  §  . 

16  fluid  ounces     "    1  pint,  O. 
8  pints  "   1  gallon,  Cong. 


APPENDIX. 


313 


478.    M^'iceUaneous. 


Blue  grass  seed 
Timothy  seed  . 
Clover  seed 
Wheat  bran     . 


14  lb.  =  1  bu. 
45  lb.  =  1  bu. 
60  lb.  =  1  bu. 
20  lb.  =  1  bu. 


Com  or  rye  meal  .   50  lb.  =  1  bu. 
Corn  or  rye      .     .    56  lb.  =  1  bu. 


Oats .  .  . 
Barley  . 
Wheat  .  . 
Beans  .  . 
Castor  beans 
Potatoes 


32  lb.  =  1  bu. 

48  lb.  =  1  bu. 
60  lb.  =  1  bu. 
60  lb.  =  1  bu. 
46  lb.  =  1  bu, 
60  lb»  =  1  bu. 


100  pounds  dry  fish         are  1  quintal, 
196  pounds  flour  "    1  barrel, 

200  pounds  beef  or  pork   "   1  barrel. 

25.  The  distance  between  two  places,  as  measured  by  a  sur* 
reyor,  is  120  chains  75  links.    How  far  are  they  apart  in  miles  ? 

26.  A  rectangular  field  is  30  chains  25  links  long,  and  25 
chains  40  links  wide.    How  many  acres  are  there  in  the  field  ? 

27.  A  piece  of  land  is  in  the  form  of  a  right-angled  triangle. 
The  sides  forming  the  right  angle  measure  14  chains  50  links 
and  24  chains  20  links.     How  many  acres  in  the  piece  ? 

28.  When  a  vessel  at  sea  has  made  500  nautical  miles,  how 
many  common  miles  has  she  gone  over  ? 

29.  How  many  prescriptions  of  20  grains  each  can  be  put 
up  from  1  lb  8  i  2  3? 

30.  How  many  minims  are  there  in  12  fluid  ounces  ? 

31.  What  is  the  value  of  a  car-load  of  corn  weighing  20000 
pounds  at  63  cents  a  bushel  ? 

32.  How  much  is  niade  by  buying  4  barrels  of  clear  pork 
at  $  19.50  a  barrel,  and  retailing  it  at  14  cents  a  pound  ? 


GOVERNMENT    LANDS. 

479.  Government  Lands  are  divided  by  parallels  and  me- 
ridians into  Townships,  6  miles  square,  each  containing  36 
Sections,  or  square  miles.  Each  section  is  subdivided  into 
half-sections  and  quarter-sections. 


314  APPENDIX. 

480.  The  Townshijos  are  numbered  witli  reference  to  two 
special  lines,  the  one  running  east  and  west,  called  the  base 
line,  and  the  other  running  north  and  south,  called  the  prin- 
Clival  meridian.     Thus, 

A  township  in  the  fourth  tier  of  townships  north  of  the 
base  line,  and  the  second  in  that  tier  west  of  the  principal 
meridian,  would  be  designated  Township  4  iV*.,  Range  2  W. 

481.  The  Sections  into  which  a  township  is  divided  are 
designated  by  numbers  beginning  with  the  northeast  corner 
section,  and  running  westward  with  the  north  tier  of  sections, 
eastward  with  the  second  tier,  and  so  on.     Thus, 

A  section  in  the  third  tier  from  the  north  in  a  township 
and  the  fourth  toward  the  west  in  the  tier  would  be  designated 
section  16. 

33.  A  section  is  in  the  fourth  tier  from  the  north  in  a  town- 
ship and  the  third  toward  the  west  in  the  tier.  What  is  its 
number  ? 

34.  Draw  a  plan  of  a  township  laid  off  into  36  sections,  and 
designate  by  number  each  section  ? 

LONGITUDE    AND    TIME. 

482.  Longitude  is  distance  east  or  west  from  a  given 
meridian,  measured  in  degrees,  minutes,  and  seconds. 

The  Prime  Meridian  is  the  meridian  from  which  the  longi- 
tude is  measured. 

483*  When  two  places  are  on  the  same  side  of  the  prime 
meridian,  their  difference  of  longitude  is  found  hy  subtracting 
the  lesser  from  the  greater  longitude  ;  when  on  opposite  sides,  by 
adding  the  longitudes. 

Note. — If  the  sum  of  the  longitudes  exceeds  180°,  it  must  be  sub- 
tracted from  360°,  since  it  is  impossible  for  the  difference  in  the  longitude 
of  two  places  to  be  more  than  180°. 

484*  The  earth,  by  turning  upon  its  axis  once  in  24  hours, 
causes  ^  of  360"^,  or  15°  of  longitude,  to  pass  under  the  sun  in 
1  hour  (Art.  179;,  and  ^  of  15°,  or  15',  in  1  minute  of  time, 
and  ^  of  15',  or  1§",  in  1  s^^op^  9^  ^^™®* 


APPENDIX.  315 

t 

485.    TABLE. 

A  difference  of  15°  in  long,  gives  a  difference  of  1  h.  in  time. 

"  15' in  long.  "  "  1  min.  in  time. 

"  15"  in  long.  **  "  1  sec.  in  time. 

Note. — The  New  Standard  Time,  adopted  in  the  United  States,  since 
1883,  by  general  agreement,  is  based  upon  four  standard  meridians,  15° 
apart,  giving  a  difference  of  just  one  hour  in  time  from  one  meridian  to 
the  next,  and  the  time  of  each  meridian  extending  to  7|°  on  each  side 
of  it. 

The  Eastern  Standard  Time  is  based  upon  meridian  75°  west  from 
Greenwich,  England,  which  passes  near  Philadelphia;  the  Central  Standard 
Time,  upon  meridian  90°  west,  which  passes  nearly  through  New  Orleans 
and  St.  Louis;  the  Mountain  Standard  Titne,  upon  meridian  105°  west, 
which  passes  near  Pike's  Peak  and  Denver ;  and  tlie  Pacijic  Standard 
Time,  upon  meridian  120°  west,  which  passes  9^'  east  of  San  Francisco. 

486.    From  the  table  is  derived  the 

Rule. 

Divide  the  difference  of  longitude,  in  degrees,  minutes,  and 
seconds,  hy  15,  and  the  quotient  will  give  the  difference  o/ 
time  in  hours,  minutes,  and  seconds. 

Multiply  the  difference  in  time,  in  hours,  minutes,  and  sec- 
onds, hy  15,  and  the  product  will  give  the  difference  in  degrees^ 
minutes,  and  seconds. 

35.  When  it  is  noon  at  New  York,  74°  0'  3^'  W.,  what  time 
is  it  at  San  Francisco  122°  W  16"  W.  ? 

122°     26'  15''  Solution.  —  Both  cities  being 

^4°        Qf  ^if  west  of  Greenwich,  their  dif- 

■iK\~A^     ^f  ^,  ference  of  longitude   is  found 

^^^  — — — .  by  subtraction.   As  15°,  15',  15" 

3  h.  13  min.    45  sec.      •     ^^ake  a  difference  of  1  h.,  1  min., 

12  h.     0  min.       0  sec.  i  sec.  of  time,  respectively,  3^ 

8  h.  46  min.    15  sec.  of  the  difference  in  longitude 

considered   as   hours,   minutes, 

and  seconds,  or  3h.  13  min.  45  sec,  will  be  the  difference  in  time. 

San  Francisco,  being  the  more  westerly,  has  earlier  time  than  New 

York.     Hence,  when   it  is  12  M.  at  the  latter  city,  it  is  3  h.  13  min. 

45  sec,  before  12  M.  at  San  Francisco,  or  8  o'clock  46  min.  15  sec.  a,  M, 


316 


APPENDIX. 


36.  What  is  the  difference  of  time  between  Greenwich  and 
Washington,  77°  2'  48^'  west  ? 

37.  Paris  is  in  2°  20'  15^'  east  longitude,  and  Boston  Tl"" 
4'  9''  west.  When  it  is  6  o'clock  in  the  morning  in  Boston, 
what  time  is  it  in  Paris  ? 

38.  How  many  degrees  east  of  Greenwich  is  E-ome,  whose 
time  is  49  min.  48}f  sec.  later  than  that  of  Greenwich  ? 

39.  The  difference  of  time  between  Baltimore  and  New 
Orleans  is  53  min.  30  sec.    What  is  the  difference  in  longitude  ? 

40.  A  gentleman  traveling  west  from  Augusta,  Maine,  in 
longitude  69°  50'  west,  found,  on  arriving  at  Des  Moines,  that 
his  watch,  which  was  right  when  he  left  Augusta,  was  1  h. 
35  min.  20  sec.  faster  than  the  time  at  Des  Moines.  What  is 
the  longitude  of  Des  Moines  ? 


LEG-AL    INTEREST. 

487.  The  Legal  Rate  of  interest  is  the  rate  established  by 
law. 

488.  When  no  rate  is  mentioned  the  legal  rate  is  that 
given  in  the  left-hand  column ;  and  if  specified  in  writing,  any 
rate  not  exceeding  that  in  the  right-hand  column  is  legal. 

TABLE. 


state. 

Rate. 

Alabama... 

8 

8 

Arkansas 

6 

10 

California 

7 

Any 

Colorado... 

10 

Any 

Conn 

6 

6 

N.  Dakota 

7 

12 

Delaware 

6 

6 

Florida   ... 

8 

Any 

Georgia  ... 

7 

8 

Illinois    ... 

6 

8 

Indiana  ... 

6 

8 

Iowa    

6 

10 

Kansas    . . . 

7 

12 

Kentucky 

6 

8 

State. 

Rate. 

Louisiana 

5 

8 

Maine 

6 

Any 

Maryland 

6 

6 

Mass 

6 

Any 

Michigan 

7 

10 

Minnesota 

7 

10 

Mississippi 
Missouri... 

6 

10 

6 

10 

Montana 

10 

Any 

Nebraska 

7 

10 

Nevada  ... 

10 

Any 

N.  H 

6 

6 

N.J 

6 

6 

N.  Mexico 

6 

12 

State. 


N.  Y 

N.  C 

Ohio    

Penn 

R.  I 

S.  C 

Tennessee 

Texas 

S.  Dakota 
Vermont 
Virginia  .. 
Washington 
W.  Va.    .. 
Wisconsin 


Rate. 


Any 

10 

6 

12 

12 

6 

8 

Any 

6 

10 


APPENDIX.  317 

TWELVE    PER    CENT   INTEREST. 

489.  The  Twelve  per  cent  method  of  computing  interest  is 
often  the  most  convenient. 

At  12  %  1  year's  interest  =  .12  of  the  principal. 

"      "     T^  y-5  or  1  month's,  interest  =  .01       "         " 
*^      "     ^0  ^-y  or  3  days',  interest       =  .001     "         " 

490.  Hence,  to  find  interest  at  12  %, 

Multiply  .01  of  the  jpTincij^al  by  the  time  in  months. 
Or, 

Multiply  .001  of  the  principal  by  J  of  the  time  in  days, 

41.   What  is  the  interest  of  $  825  for  2  y.  7  mo.  7  d.  at  4  %  ? 
$825     =:  Principal. 


)  8.25    =  1  mo's  interest. 
31.2J^  =  Time  in  months. 


Solution.  —  1  month's  in- 
terest at  12%  is  yf^  of  the 
^'^^  principal,   or   $8.25;    31.2J 

1650  months'  interest  is  31.2 J  X 

825  $8.25,   or    $257,675.      The 

2475  interest  at  4%  =  ^  of  12% 


3)  $257.675    =  12  %  interest.  interest,  or  $  85.89. 

$85.89      ==4%  interest. 
42.   Find  the  interest  of  $  840  for  73  days  at  10  %. 
$  840    =  Principal. 
$  0.84    =  3  days'  interest. 
24^  =  \  time  in  days. 

336 

168 
12)$  20.44    =  12%  interest. 
$  1.703  =  1  %  interest. 
$  17.03  =  10  %  interest. 


318  APPENDIX. 

491.     Connecticut  Rule  for  Partial  Payments. 

TVhen  at  least  a  yeai^^s  interest  has  accrued  at  the  time  of  a 
payment,  and  in  the  case  of  the  last  payment,  follow  the  Uiiited 
States  Rule. 

When  less  than  a  year's  interest  has  accrued  at  the  time  of  a 
payment,  except  the  last,  find  the  difference  between  the  amount 
of  the  principal  for  an  entire  year,  and  the  amount  of  the  pay^ 
TTient  for  the  remainder  of  the  year  after  it  is  vfiade,  for  a  new 
^principal. 

When  the  interest  which  has  accrued  at  the  time  of  a  pay- 
ment exceeds  the  payment^  find  the  interest  upon  the  principal 
only. 

Note. — At  the  option  of  the  teacher,  the  exercises  under  the  United  Sf^ates 
Rule  (Art.  282),  may  be  performed  by  this  rule. 

ANNUAL  INTEREST. 

492.    Annual  Interest  is  interest  payable  annually. 
The  annual  interest  not  paid  when  due  draws  simple  intei 
est. 

43.  What  is  the  amount  due  on  a  note  of  %  500,  interest 
payable  annually,  on  which  no  payments  have  been  made,  at 
the  end  of  3  years  6  months  and  12  days  ? 

Solution. 

Principal $500.00 

Ist  annual  interest $30.00 

Int.  on  1st  annual  int.  2  y.  6  mo.  12  d.    .  $  4.56 

2d  annual  interest 30.00 

[nt.  on  2d  annual  int.  1  y.  6  mo.  12  d.      .  2.76 

3d  annual  interest 30.00 

Int.  on  3d  annual  int.  6  mo.  12  d.      .     .  .96 

4th  annual  interest 16.00 

S  500.00  $106.00     $8.28 

Total  interest,  $  106  +  $  8.28  =  114.28 

Amount  due $  614.28 


APPENDIX.  319 

493.     Rule  for  Annual  Interest. 

Compute  the  interest  annually,  and  simple  interest  on  each 
annual  interest  for  the  time  it  shall  remain  unpaid, 

44.  A  debt  of  I  600  was  contracted  Jan.  1,  1880 ;  allowing 
interest  annually  at  6%,  and  if  no  payments  he  made,  what 
will  be  due  April  1,  1884  ? 

45.  A  note  was  given  May  16,  1881,  for  $  1250,  interest  an- 
nually at  5  % ;  if  no  payment  is  made,  what  will  be  due  March 
^.6,  1884  ? 

46.  A  note  was  given  March  14,  1880,  for  $  576,  interest 
annually  at  6  %.  What  will  be  due  Sept.  26,  1883,  no  pay- 
ments having  been  made  ? 

494.  When  Partial  Payments  have  been  made  on  a  note, 
or  other  obligation  drawing  annual  interest,  the  following  is 

The  New  Hampshire  Rule. 

If  in  any  year,  reckoning  from  the  time  the  annual  interest 
began  to  accrue^  payments  have  been  made,  compute  interest 
upon  them  to  the  end  of  the  year  in  which  they  are  made. 

The  amount  of  payments  is  to  be  then  applied,  first,  to 
cancel  interest  upon  annual  interest ;  second,  to  cancel  annual 
interest ;  and  thirdly,  to  the  extinguishment  of  the  principal. 

If  however,  at  the  date  of  any  payment  there  is  no  interest 
except  the  accruing  annual  interest,  and  the  payment,  or  pay- 
ments, do  not  exceed  the  annual  interest  at  the  end  of  the  year, 
deduct  the  payment,  or  payments,  without  interest  on  the  same. 

Omit  the  last  paragraph  of  this  rule  and  it  is  the  Vermont  Rule. 

47.  What  was  due,  July  1,  1881,  on  a  note  dated  July  1, 
1878,  for  %  1000,  with  6  %  annual  interest,  and  on  which  was 
paid,  Dec.  1,1879,  $400? 


320  APPENDIX. 

Solution. 

Principal $1000 

1st  annual  interest $60.00 

Int.  on  1st  annual  int.  2  y $3.60 

2d  annual  interest 60.00  

$1000         $120.00  $3.60 

Payment  Dec.  1, 1879  .     .$400.00 

Int.  on  paym't,  July  1,  1880     14.00 

AnVt  of  payin't  July  1, 1880  $414.00  =  290.40   +    120.00  +        3.60 

Principal,  July  1, 1880,  .     .  $  709.60 

3d  annual  interest     .     .     .  $42.58 

Due  July  1,  1881   .     .  $709.60  +  $42.58  =  $  752.18 

48. 

$2000.  Manchester,  N.  H.,  April  1,  1873. 

On  demand,  I  promise  to  pay  Charles  West  &  Son,  two 
thousand  dollars,  value  received,  with  interest  annually. 

James  Goddard. 

Payments:  Sept.  19,  1875,  $500;  Dec.  3,  1879,  $600;  and 
Aug.  9,  1880,  $  775.     What  will  be  due  May  19,  1883  ? 

49. 

$5000.  Plymouth,  N.  H.,  Jan.  13,  1874. 

On  demand,  we  promise  to  pay  to  the  order  of  John  M. 
Monroe  &  Co.,  five  thousand  dollars,  value  received,  with  in- 
terest annually.  Presby  &  Lord. 

Payments :  Sept.  23,  1878,  $  2000 ;  Feb.  19,  1880,  $  1500  ; 
May  29,  1881,  $  125  ;  and  June  11,  1883,  $  20.  What  will  be 
due  at  settlement,  Aug.  30,  1885  ? 

Note.  —  At  the  option  of  the  teacher,  each  of  the  above  exercises  may  be 
i^erfonned  by  the  Vermont  Rule. 


APPENDIX. 


&21 


AVERAGE   OF  ACCOUNTS. 

495.  The  Balance  of  an  account  is  the  difference  between 
its  debtor  and  creditor  sides. 

496.  The  Average  of  an  account  is  the  equitable  time  of 
the  balance  becoming  due,  or  being  entitled  to  interest. 

50.    What  is  the  balance  of  the  following  account,  and  at 
what  date  should  the  balance  begin  to  draw  interest  ? 


Pr. 

JOHN  L. 

MARTIN. 

(Er. 

1880. 

1880. 

May  16 

Mdse. 

on 

60  d. 

1300 

May  20 

Mdse. 

on 

30  d. 

$200 

June   3 

u 

u 

60  d. 

50 

July  19 

u 

a 

60  d. 

200 

July    1 

u 

a 

30  d. 

150 

DeMts. 

July  15,  300  X  26  d.  =  7800  d. 

Aug.  2,   50  X  44  d.  =  2200  d. 

July  31,     150  X  42  d.  —  6300  d. 

$500  16300  d. 

400 
$  100  balance. 


Solution. 

Credits. 
June  19,  200  X    0  d.  : 
Sept.  17,  200  X  90  d.  : 

$400 


Od 
18000  d. 
18000  d. 
16300  d. 


1700 -f-  100  =  17  d. 


1700  d- 


June  19  —  17  days  =  June  2,  average  time. 

We  select  for  convenience  June  19,  the  earliest  date  at  which  any 
of  the  items  of  account  become  due,  as  the  point  of  reckoning,  and 
find  the  aggregate  of  the  terms  of  credit  of  the  credit  items,  with  ref- 
erence to  the  selected  date,  to  be  equal  to  the  credit  of  $  1  for  18000 
days,  and  the  aggregate  of  the  terms  of  credit  of  the  debit  items  to  be 
equal  to  the  credit  of  $  1  for  16300  days. 

Striking  the  balance,  it  appears,  at  the  selected  date,  $  100  subject 
to  a  credit  equal  to  the  credit  of  f  1  for  1700  days  is  in  favor  of  John  L. 


322 


APPENDIX. 


Martin.  But  the  credit  of  $  1  for  1700  days  is  equal  to  that  of  $  100 
for  Yoo  of  1700  days,  or  17  days;  hence  the  $  100  was  due  in  equity 
17  days  before  June  19,  or  June  2. 

If,  however,  the  balance  of  items  and  terms  of  credit  had  been  both 
on  the  same  side  of  the  account,  the  balance  would  have  been  due  af- 
ter, instead  of  before,  the  selected  date. 

497.     Rule  for  Averaging  Accounts. 

Select  the  earliest  date  at  which  any  of  the  items  of  account 
becomes  due,  and  therefrom  reckon  the  terms  of  credit. 

Multiply  each  term  of  credit  hy  the  number  denoting  the  cor- 
responding item,  and  divide  the  balance  of  the  sums  of  the 
'products  by  the  balance  of  the  sums  of  the  items  of  the  account, 
and  the  quotient  will  be  the  average  term  of  credit. 

When  the  balances  are  both  on  the  same  side  of  the  account,  the 
time  must  be  added  to  the  selected  date,  but  subtracted  from  that  date 
when  the  balances  are  on  different  sides. 

The  same  result  may  be  reached  by  what  is  called  the  Interest 
Method.     Thus, 

Compute  interest  upon  each  item  for  the  days  intervening^  between  its 
becoming  due  and  the  earliest  date  at  which  any  item  becomes  due.  Di- 
vide then  the  balance  of  interest  by  the  interest  of  the  balance  of  items  for 
QTie  day,  and  the  quotient  will  be  the  average  term  of  credit. 

Note  1.  —  A  convenient  rate  of  interest  in  averaging  accounts  is  12  %,  which 
is  .01  of  the  principal  for  30  days  and  .001  for  3  days. 

Note  2.  —  The  note  under  the  rule  Art.  334  applies  in  averaging  accounts. 

51.  Find  when  the  balance  of  the  following  account  averages 
due. 


Pr. 


WILLIAM  HOLT. 


dr. 


1881. 
June  20 
July  10 


Mdse.  on  30d. 
"     net 

$600.60 
149.40 

1881. 
July  15 

Cash 


$650 


APPENDIX. 


323 


52.  Find  the  balance  of  the  following  account,  and,  allow- 
ing each  item  to  be  on  30  days,  the  time  the  balance  becomes 
due. 


Pr. 

BRYANT  AND  ROBERTS. 

&X. 

1881. 
Sept.  30 
Oct.    15 

Mdse. 

11 

$550 
850 

1881. 

Oct.  1 

"      5 

Mdse. 

$400 
30 

53.   Find  the  time  when  a  note  for  the  balance  of  the  follow- 
ing equated  account  should  begin  to  draw  interest. 


Due  March  2 


Dr. 

$600 


Due  March  14 


Cr. 

$400 


54.  Find  the  face  of  a  note  which  must  be  given  for  the  bal- 
ance of  the  following  account,  and  the  date  at  which  it  should 
begin  to  draw  interest. 

pr.  JAMES  GRIMSHAW.  (Ir. 


1881. 

1881. 

Aug.  10 

"     28 

Mdse.  on  4  mo. 
"       "   6  mo. 

$200 
200 

Sept.  11 
Dec.   10 

1882. 

Cash. 
it 

$60 
140 

Sept.    1 

"      net 

150 

Jan.  29 

.    u 

100 

BUSINESS    FORMS. 
RECEIPTS. 

498.    A  Receipt  is  a  written  acknowledgment  that  money 
or  other  property  of  value  has  been  received. 

Receipt  for  Payment  on  Account 

■  $  250.  Boston,  Oct.  19,  1881. 

Received  from  Henry  F.  Jordan  Two  hundred  fifty  dol- 
lars on  account.  Smith  &  Watson, 


324  APPENDIX. 

Receipt  for  Rent. 

Received,  Portland ,  May  5,  1882,  from  James  Johnson^ 
Thirty-one  dollars  for  rent  of  hoicse,  No.  14  Austin  Street, 
for  month  ending  April  30,  1882. 
$  31.  Catharine  A,  Pettes. 

Receipt  in  Full. 

$  26-Mr.  Philadelphia,  Aug.  1,  1882. 

Received  from  John  Randall  Twenty-six  -^^q  Dollars  in 

full  of  all  demands  to  date. 

S.  A.  Caswell  &  Co. 

ORDERS. 

499.  An  Order  is  a  written  request  to  deliver  money  or 
goods  to  some  person  mentioned,  or  to  his  order,  or  to  the 
bearer,  on  account  of  the  person  signing  the  request. 

Order  for  Money. 

Burlington,  Dec.  12,  1881. 
Mess7's.  Simmons  &  Son, 

Gentlemen :  Please  jpay  S.  P.  Wright,  or  order,  Seventy- 
five  Dollars,  and  charge  to  our  account. 

Reed,  Pratt,  &  Co. 

Order  for  Goods. 

Manchester,  Nov.  4,  1881. 
Mr.  W.  N.  Goddard, 

Please  pay  to  James  Brewer,  or  order,  Sixty  Dollars  in 

Goods  from  your  store,  and  charge  to  the  account  of 

Charles  Dole. 

DUE-BILLS. 

500.  A  Due-Bill  is  a  simple  acknowledgment  of  a  debt 
in  writing. 


APPENDIX. 


325 


Due-Bill  for  Goods. 

Due,  New  York,  Aug.  9,  1882,  to  H.  L.  Chase,  Twenty- 
three  -^^Q  Dollars  in  goods  from  my  store. 
$  23^^.  Henry  (7.  Carter, 

CHECKS. 

501.  A  Check  is  a  written  order  addressed  to  a  bank  by  a 
person  having  money  deposited,  requesting  the  payment  on 
presentation  of  a  certain  sum  of  money  to  a  person  named 
therein,  or  to  his  order. 

Bank  Check. 


The  National  Bank  of  Commerce. 


S^.    ^lOT^nc'na,  Oi   Oidci, 


Scad^n-aezf^en . 


/^  ^o//aU. 


TAXES. 

502.  A  Tax  is  a  sum  of  money  assessed  upon  the  person 
property,  income,  or  business  of  individuals  for  public  use. 

503.  A  Poll  Tax  is  a  tax  upon  the  person,  a  Property  Tax 
is  a  tax  on  property,  and  an  Income  Tax  is  a  tax  on  income. 

504.  Assessors  are  officers  appointed  to  take  an  inventory 
of  taxables,  and  to  apportion  the  tax  to  be  raised  among  the 
tax-payers. 


326 


APPENDIX. 


55.  A  town  is  to  be  taxed  $  14562.  The  taxable  property 
is  $1146000.  There  are  540  polls,  each  assessed  $1.50. 
What  will  be  A's  tax,  whose  property  is  assessed  at  $  8500, 
and  who  pays  one  poll-tax  ? 

Solution. 

$  1.50  X  540  =  $  810,  sum  to  be  assessed  on  the  polls. 

$  14562  —  $  810         =  $  13752,  sum  to  be  assessed  on  the  property. 

$  13752  -f-  ^  1146000  =  0.012,  or  12  mills  on  $  1  of  valuation. 

$  8500  X  0.012  =  $  102,  As  tax  on  property. 

a  102  +  $  1.50  =  $  103.50,  As  entire  tax. 

505.    Rule  for  Assessment  of  Taxes. 

Deduct  the  amount  of  the  poll-taxes,  if  any,  from  the  entire 
tax  to  he  raised,  and  the  remainder,  divided  by  the  value  of 
the  taxable  property,  will  give  the  rate. 

Multiply  the  value  of  each  individuaVs  taxable  property  by 
the  rate,  and  to  the  product  add  the  poll-tax,  if  any,  and  the 
sum  will  be  the  individuaVs  entire  tax. 

Note.  —  In  Massachusetts,  the  assessors  "in  each  year  assess  upon  the  polls 
the  state  and  county  taxes  authorized  or  required  by  law  ;  provided,  however, 
that  in  case  either  of  said  taxes  shall  exceed  in  amount  the  sum  of  one  dollar 
upon  each  poll,  the  excess  above  said  amount  and  in  every  case  the  whole  amount 
assessed  for  other  purposes  shall  be  apportioned  upon  property.  .  .  .  The  state 
tax  assessed  upon  poll  and  property  and  the  county  tax  assessed  upon  poll  and 
property  shall  each  constitute  an  entire  and  indivisible  tax."  —  CAop.  299  o/ 
the  Acts  of  1879. 

Computation  of  taxes  may  be  facilitated  by  the  construction  of  a  table. 
Thus,  if  the  rate  on  $  1  is  12  mills,  we  can  have  the  following 

TABLE. 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$0,012 

$8 

$0,096 

$60 

$0.72 

$400 

$4.80 

$2000 

$24.00 

2 

0.024 

9 

0.108 

70 

0.84 

500 

6.00 

3000 

36.00 

3 

O.O.'iG 

10 

0.12 

80 

0.96 

600 

7.20 

4000 

48.00 

4 

0.048 

20 

0.24 

90 

1.08 

700 

8.40 

5000 

60.00 

5 

0.060 

30 

0.36 

100 

1.20 

800 

9.60 

6000 

72.00 

6 

0.072 

40 

0.48 

200 

2.40 

900 

10.80 

7000 

84.00 

7 

0.084 

50 

0.60 

300 

3.60 

1000 

12.00 

8000 

96.00 

APPENDIX.  327 

56.  What  is  B's  tax,  by  the  table,  his  valuation  being 
$  2545,  and  he  paying  one  poll-tax  of  $  1.75  ? 

57.  Find,  by  the  table,  C's  tax  on  $  9565,  D's  on  $  1764, 
and  E's  on  $  5630,  and  each  paying  a  poll-tax  of  $  1.50. 

58.  The  taxable  property  of  a  certain  town  is  $  1000000. 
The  number  of  polls,  600.  The  tax  to  be  raised  is.  State 
$348,  County  $1500,  and  Town  $12100.  The  state  and 
county  tax  are  each  to  be  assessed  upon  the  polls,  but  so  much 
as  either  of  these  taxes  shall  exceed  $  1  on  a  poll,  is,  with  the 
town  tax,  to  be  assessed  upon  the  property.  What  will  be 
each  poll-tax  ?  What  will  be  the  rate  of  county  tax  ?  What 
will  be  the  rate  of  town  tax  ?  What  will  be  A's  county  tax, 
his  valuation  being  $  5000,  and  he  paying  for  one  poll  ?  What 
will  be  his  entire  tax  ? 

DUTIES,    OR    CUSTOMS. 

506.  Duties,  or  Customs,  are  taxes  levied  on  imported 
goods  and  the  tonnage  of  vessels. 

They  are  collected  at  custom-houses  by  the  government 
officers  in  charge. 

507.  A  Specific  Duty  is  a  fixed  tax  upon  an  article  with- 
out regard  to  its  value. 

508.  An  Ad  Valorem  Duty  is  a  tax  at  a  certain  rate  on 
the  cost  of  the  goods  in  the  country  from  which  they  are 
imported. 

509.  Tare  is  an  allowance  made  for  the  weight  of  the  box, 
cask,  etc.,  containing  the  goods. 

It  may  be  estimated  by  actual  weighing,  or  by  a  schedule 
furnished  by  the  government. 

Leakage  is  an  allowance  for  waste  of  liquors  in  casks,  and 
Breakage  is  an  allowance  on  liquors  in  bottles. 

Gross  Weight  is  the  weight  before  any  allowances  are  made, 
and  Net  Weight  is  the  weight  after  the  allowances  are  made. 


328 


APPENDIX. 


610.  Values  of  Foreign  Coins,  in  United  States  money, 
to  be  followed  in  estimating  values  of  foreign  merchandise  at 
custom  houses,  as  proclaimed  by  the  Secretary  of  the  Treasury, 
January  1,  1890. 

TABLE. 


Country. 

Standard. 

Monetary  Unit. 

Value   in 

terms  of 

U.  S.  Gold 

dollar. 

Argentine  Repub. 

Austria 

Brazil 

British  Poss.  N.  A. 

Chili 

Cuba 

Egypt 

France 

German  Empire. . 
Great  Britain .... 
India 

Japan 

Mexico 

Netherlands 

Norway 

Portngal 

Russia 

Tripoli 

Turkey 

Colombia 

Venezuela 

Gold  and  silver 

Silver 

Gold 

Gold 

Gold  and  silver 
Gold  and  silver 

Gold 

Gold  and  silver 

Gold 

Gold 

Silver 

Gold  and  silver 

Silver 

Gold  and  silver 

Gold 

Gold 

Silver 

Silver 

Gold 

Silver 

Silver 

Peso 

Florin 

Milreis  of  1000  reis... 

Dollar 

Peso 

Peso 

Pound  (100  piasters). 

Franc 

Mark 

Pound  Sterling 

Rupee  of  16  annas. . . 
(Gold 

$0,965 
0.345 
0.546 
1.00 
0.912 
0.926 
4.943 
0.193 
0.238 
4.866^ 
0.332 
0.997 
0.752 
0.758 
0.402 
0.268 
1.08 
0.558 
0.629 
0.044 
0.698 
0.14 

Y^^  ]  Silver. .... .... 

Dollar 

Florin 

Crown 

Milreis  of  1000  reis. .. 
Rouble  of  100  copecks 
Mahbub  of  20  piasters 

Piaster 

Peso 

Bolivar 

Note.  The  par  of  exchange  of  the  monetary  unit  of  a  country  with  a  gold,  and  gold  and 
silver,  standard  is  fixed  at  the  value  of  the  gold  unit  as  compared  with  the  United  State? 
gold  unit;  and,  in  case  of  a  single  silver  standard,  the  par  is  computed  at  the  mean 
price  of  silver  in  the  London  market,  from  October  1,  to  December  24,  1889. 

The  boliviano  of  Bolivia,  the  Sucre  of  Ecuador,  the  peso  of  Guatemala,  Honduras, 
Nicaragua,  Salvador,  and  Port  a  Rico,  and  sol  of  Peru^  silver  standard,  are  each  of  the 
same  value  as  the  peso  oiColombia. 

The  franc  of  Belgium,  the  franc  of  Switzerland,  the  pesata  of  Spain^  the  lira  of  Italy, 
and  the  drachma  of  Greece,  gold  and  silver,  are  each  of  the  same  value  as  the  franc  of 
France. 

The  crown  of  Sweden,  and  the  crown  of  Denmark,  gold,  are  each  of  the  same  value 
as  the  crown  of  Norway. 

59.  The  Bay  State  Iron  Co.  imported  from  England  200  tons 
pig-iron,  invoiced  at  £725  35.  4cZ.,  and  paid  a  duty  of  $7  per  ton. 
Find  the  duty  and  the  cost  of  the  importation. 


APPENDIX.  329 

60.  Jordan,  Marsh,  &  Co.  import  from  Paris  3  cases  silk 
goods  containing  2664.5  meters,  invoiced  at  6  francs  per  meter. 
They  pay  a  duty  of  60  %.    What  does  the  government  receive  ? 

61.  What  is  the  duty  on  an  importation  from  Russia  of  516 
bales  of  flax,  weighing  7880  poods,  at  $20  per  ton,  a  pood 
being  a  Russian  weight  of  about  36  lb.  ? 

62.  Spaulding,  Nash,  S^Co.  receive  from  Cuba  an  invoice 
of  50381  gallons  of  molasses,  valued  at  11102.7  pesos.  They 
pay  a  duty  of  5/  per  gallon  and  25%  ad  valorem.  How  many 
dollars  do  they  pay  the  collector  of  customs  ? 

63.  R.  H.  White  &  Co.  import  from  Germany  a  case  of 
dress-goods,  containing  640.4  meters,  costing  in  Hamburg 
2.05  marks  per  meter  less  a  discount  of  8  %.  The  width  of 
the  goods  is  43|  inches.  The  duty  paid  is  8  /  per  square  yard 
and  40  %  ad  valorem.     Find  it. 

MEASUREMENT    OF    ROUND    TIMBER. 

511.  Bound  Timber  (or  logs)  is  usually  estimated  in  cubic 
feet. 

512.  Spars  from  10  to  4  inches  in  diameter,  inclusive,  are 
estimated  by  the  inch  diameter,  taken  clear  of  bark,  at  one  third 
of  their  length  from  the  larger  end. 

Spars  above  7  inches  should  have  4  feet  of  length,  and  below  7 
inches  should  have  5  feet  of  length,  to  every  inch  of  diameter. 

513.  The  Mean  Girt  of  a  tapering  piece  of  round  timber  is 
the  girt,  clear  of  bark,  at  one  third  its  length  from  the  larger 
end. 

514.  To  find  the  contents  of  round  timber  in  cubic  feet. 

Multiply  the  length  in  feet  hy  the  square  of  one  fourth  of 
the  mean  girt  in  inches^  and  divide  the  product  hy  144. 

Note.  —  The  rule  gives  about  a  fifth  less  than  the  exact  quantity,  so  much 
l»eing  allowed  for  crooks  and  waste. 


330  APPENDIX. 

64.  How  many  cubic  feet  of  timber  in  a  log  whose  length  is 
30  feet,  and  whose  mean  girt  is  42  inches  ? 

65.  The  mean  girt  of  a  piece  of  round  timber  is  60  inches, 
and  its  length  24  feet.     Required  its  contents  in  cubic  feet. 

515.  To  find  the  side  of  squared  timber  that  may  be  hewr 
Dr  sawed  from  a  log, 

Multiply  the  diameter  of  the  smaller  end  of  the  log  by 
0.707. 

66.  The  smaller  end  of  a  log  is  21  inches  in  diameter.  What 
is  the  side  of  the  squared  beam  that  may  be  sawed  from  it  ? 

67.  How  many  board  feet  in  a  log  when  squared,  the  length 
of  the  log  being  18  feet,  and  its  diameter  at  the  smaller  end 
24  inches  ? 

68.  A  log  is  in  the  form  of  a  cylinder,  6  feet  in  circumfer- 
ence and  20  feet  long.  What  is  the  value  of  the  largest  squared 
timber  that  can  be  hewn  from  it,  at  $  30  a  thousand  feet,  board 
measure  ? 

GAUGING. 

516.  Gauging  is  finding  the  capacity  of  casks. 

517.  The  Mean  Diameter  of  a  cask  is  very  nearly  equal  to 
the  head  diameter  increased  by  0.55  to  0.70  of  the  difference 
between  the  bung  and  head  diameters,  acrording  as  the  staves 
are  curved  little  or  much. 

518.  To  find  the  capacity  of  casks, 

Multiply  the  product  of  the  square  of  the  mean  diameter 
and  the  lengtli,  expressed  in  inches^  by  0.0034  for  gallons^  or 
by  0.0129  for  liters. 

As  a  cubic  foot  is  about  1\  gallons,  in  finding  the  capacity  of  a  cis- 
tern it  is  sufficiently  accurate  to  estimate  7^  gallons  to  a  cubic  foot. 


APPENDIX.  331 

69.  A.  cask  whose  mean  diameter  is  22  inches  and  length 
30  inches  will  contain  how  many  gallons  ? 

70.  What  is  the  capacity  of  a  cask  in  gallons  whose  mean 
diameter  is  30  inches  and  length  38  inches  ? 

71.  What  is  the  capacity  in  liters  of  a  cask,  staves  much 
curved,  whose  head  diameter  is  24,  bung  diameter  30,  and 
length  36  inches  ? 

72.  How  many  gallons  in  capacity  is  a  rectangular  cistern 
whose  inside  dimensions  are  4  feet  3  inches,  3  feet  6  inches, 
and  4  feet  ? 

TONNAGE  OF  VESSELS. 

519.  The  Tonnage  of  a  vessel  is  the  number  of  tons'  burdeD 
it  will  carry. 

520.  Shipwrights  generally  make  their  estimates  of  ton- 
nage  by  the  following 

Rule. 

For  a  single-deck  vessel^  take  the  length  in  feet  above  the 
deck  from  the  forepart  of  the  mainstem  to  the  after-part  of  the 
sternpost,  the  breadth  at  the  widest  part  above  the  main  wales 
on  the  outside,  and  the  depth  from  the  under  side  of  the  deck 
plank  to  the  ceiling  of  the  hold.  From  the  length  subtract 
three  fifths  of  the  breadth,  multiply  the  remainder,  breadth 
and  depth,  together,  and  the  product  divided-by  95  will  give 
the  tonnage. 

For  a  double-deck  vessel,  take  the  length  above  the  upper 
deck,  for  the  depth  half  the  breadth,  and  proceed  as  before, 

73.  What  is  the  tonnage  of  a  single-decked  vessel  whose 
length  is  75  feet,  breadth  20  feet,  and  depth  9  feet  ? 

74.  What  is  the  tonnage  of  a  double-decked  vessel  whose 
length  is  160  feet  and  breadth  30  feet  ? 


332  APPENDIX. 

FARMERS'  ESTIMATES. 

521.  Gram  in  a  bin  or  granary  occupies  nearly  1:^  as  many 
cubic  feet  as  there  are  bushels. 

522.  Corn  on  the  ear  will  yield  about  half  its  bulk  in 
shelled  corn. 

523.  Wheat,  according  to  quality,  less  a  sixth  for  toll,  will 
yield  from  26  to  33  pounds  of  flour  per  bushel. 

524.  Mixed  Hay,  in  large  mow,  is  estimated  at  500  cubic 
feet,  and  Clover,  at  550  cubic  feet  to  a  ton  of  2000  pounds. 

525.  Horses,  young  cattle,  and  sheep  are  estimated  to  con- 
sume daily,  for  each  100  pounds  of  weight,  about  3  pounds  of 
hay ;  and  oxen  and  cows,  about  2 J  pounds. 

As  food  for  stock,  100  pounds  of  average  meadow  hay  is  equal  to 
about  56  pounds  of  corn,  56  pounds  of  wheat  middlings,  60  pounds  of 
oats,  or  32  pounds  of  cotton  seed  meal. 

526.  Net  Weight  of  fat  beeves  is  about  f  of  the  live  weight ; 
of  fat  swine,  J  ;  of  fat  sheep,  J ;  and  of  fat  fowl,  /^. 

Average  beeves,  net  weight,  will  cut  up  :  rump  and  sirloin,  ^ ; 
thigh  and  round,  \  ;  forequarter  and  rattlerand,  f  ;  hide,  ^^^ ;  and 
tallow,  ^^. 

An  average  swine,  net  weight,  will  cut  up  :  hams  and  shoulders,  ^^ 
and  sides  and  clear  pork,  ^. 

75.  I  have  a  bin  8  feet  long,  4  feet  wide,  and  3  feet  deep. 
How  many  bushels  will  it  contain  ? 

76.  A  wagon  8  feet  long,  3|-  feet  wide,  and  2  feet  deep,  is 
filled  with  corn  in  the  ear.  How  many  bushels  of  shelled  corn 
will  it  yield  ? 

77.  How  much  average  meadow  hay  will  suffice  to  keep  3 
horses  120  days,  whose  weight  is  1200  pounds  each,  provided 
grain  is  allowed  to  replace  one  third  of  the  hay  ? 


APPENDIX.  333 

78.  How  many  bushels  of  best  wheat  must  be  carried  to 
mill  to  get  back,  after  allowing  a  sixth  for  toll,  a  barrel  of 
flour  ? 

79.  When  corn  is  75  cents  a  bushel,  what  is  the  correspond- 
ing value  of  average  meadow  hay  as  food  for  stock  ? 

80.  The  live  weight  of  a  fat  ox  is  1550  pounds.  If  slaugh 
fcered,  how  many  pounds  of  his  net  weight  will  cut  up  into 
rump  and  sirloin,  and  how  many  into  round  ? 

81.  The  live  weight  of  5  fat  swine  is  2250  pounds.  How 
many  pounds  of  the  net  weight  will  cut  up  into  hams  and 
shoulders,  and  how  many  into  sides,  or  clear  pork  ? 

STONE  AND   BRICK  WORK. 

527.  A  Perch  of  stone  or  masonry  is  16J  feet  long,  1^  feet 
thick,  and  1  foot  high,  or  24 J  cubic  feet. 

528.  In  Rubble  masonry,  a  cubic  yard  laid  in  the  wall  re- 
quires IJ  cubic  yards  of  undressed  stone,  and  ;|^  of  a  cubic  yard 
of  mortar. 

In  Ashlar  work,  about  ^  of  the  volume  of  the  stone  is  al- 
lowed for  mortar. 

A  mason,  with  a  helper,  can  in  a  day  lay  in  courses  4  cubic  yards 
of  rubble  stone  dry,  or  3  cubic  yards  in  mortar. 

529.  Bricks  when  laid  will  average  for  each  square  foot  of 
surface  on  the  face  of  the  wall  about  twice  as  many  bricks  ip 
number  as  the  wall  is  inches  thick. 

530.  A  Cask  of  Lime  is  about  2  J  bushels,  or  240  pounds, 
and  absorbs  about  2 J  times  its  bulk,  or  2|  times  its  weight,  of 
water  in  slacking. 

A  cask  of  lime,  with  about  10  bushels  of  sharp  sand,  will 
make  mortar  for  laying  about  1000  bricks,  or,  with  the  addi- 
tion of  5  pounds  of  hair,  mortar  for  35  square  yards  of  plaster- 
ing, one  coat  work,  or  30  square  yards,  two  coat  work  slipped. 


334  APPENDIX. 

531.  A  Cask  of  Cement  of  300  pounds,  with  twice  its  bulk 
of  sharp  sand,  will  make  mortar  for  laying  650  bricks ;  or, 
with  four  times  its  bulk,  or  about  12  bushels,  of  clean  gravel, 
concrete  for  9  square  yards  of  flooring  surface. 

532.  In  Paving,  about  40  bricks  laid  flatwise,  or  75  bricks 
laid  edgewise,  are  allowed  for  one  square  yard. 

A  mason,  with  a  helper,  in  a  day  can  lay  in  mortar,  8-inch  work, 
1400  bricks,  or  12-inch  work,  2000  bricks  ;  and  bricks  flat  in  sand  20 
Bquare  yards,  or  in  cement,  12  square  yards. 

82.  What  will  be  the  cost  of  the  material  for  mortar  for 
plastering  900  square  feet,  the  price  of  lime  being  $  0.90  per 
cask,  sand  8  cents  per  bushel,  and  hair  6  cents  per  pound  ? 

83.  I  have  a  walk  4  feet  wide  and  224  feet  long.  What  will 
it  cost  to  pave  it  with  brick,  laid  flat  in  sand,  brick  $  7.50  a 
thousand,  wages  of  the  mason  $  2.75  per  day,  and  of  a  helper 
1 1.50  per  day  ? 

84.  What  will  it  cost  to  concrete  the  bottom  of  a  cellar  40 
feet  long  and  24  feet  wide,  cement  being  $  2  a  barrel  and  gravel 
8  cents  a  bushel  ? 

85.  What  will  it  cost  to  build  a  12-inch  brick  wall,  6  feet 
high  and  100  feet  long,  laid  in  mortar,  brick  being  $  8  per 
thousand,  lime  $  1.10  per  cask,  sand  10  cents  per  bushel,  wages 
of  the  mason  $  3  per  day,  and  of  the  helper  $  1.75  per  day  ? 

86.  A  cellar  is  34  feet  long,  27  feet  wide,  and  9  feet  deep. 
Its  walls  are  IJ  feet  thick,  made  of  rubble  stones  laid  in  mor- 
tar. The  stone  undressed  cost  $  2.50  per  perch,  the  lime 
cost  $  1  per  cask,  the  sand  used  with  it  for  mortar  10  cents  a 
bushel,  and  a  cask  of  lime  and  10  bushels  of  sand  made  15  cu- 
bic feet  of  mortar.  The  mason  who  built  the  walls  was  paid 
1 3  a  day,  and  his  helper  $  2.  What  did  the  material  and 
mason  work  of  the  cellar  cost  ? 


APPENDIX.  *  335 


BUILDERS'  ESTIMATES. 

533.  Shingles  are  usually  16  inches  long,  and  on  an  aver- 
age 4  inches  wide,  and  are  put  up  4  bundles  to  the  1000. 

1000  shingles,  laid  4  inches  to  the  weather,  will  cover  107  square 
feet ;  laid  4J-  inches  to  the  weather,  120  square  feet ;  and  laid  5  inches 
to  the  weather,  133  square  feet. 

534.  Clapboards  are  usually  4  feet  long,  and  put  up  40 
bundles  to  the  1000. 

100  of  4-foot  clapboards,  laid  4  inches  to  the  weather,  will  cover 
130  square  feet ;  laid  4|-  inches  to  the  weather,  150  square  feet ;  and 
laid  5  inches  to  the  weather,  165  feet. 

535.  Laths  are  usually  4  feet  in  length,  and  are  put  up  10 
bundles  to  a  1000. 

loo  laths,  set  J  of  an  inch  apart,  will  cover  6^  square  yards. 

A  workman  in  a  day  will  set  of  laths  about  100  square  yards,  lay 
of  shingles  on  a  roof  about  2000,  or  put  on  of  outside  boards  abou^ 
1000  feet. 

536.  Nails  are  put  up  100  pounds  to  a  keg. 

6  pounds  of  4-penny,  or  5  pounds  of  3-penny,  nails  are  allowed  for 
laying  1000  shingles  ;  3^  to  4^  pounds  of  5-penny  nails  for  laying 
1000  clapboards ;  7  pounds  of  3-penny  nails  for  setting  1000  laths. 

537.  Paint  for  outside  work  may  have  for  first  coat  16J 
pounds  of  white  lead,  ground  in  oil,  to  a  gallon  of  linseed  oil ; 
and  for  second  or  third  coat,  20  pounds  of  white  lead  to  a  gal- 
lon of  linseed  oil. 

For  inside  work,  the  spirits  of  turpentine  may  replace  from  one 
third  to  two  thirds  of  the  oil. 

538.  A  Gallon  of  linseed  oil  weighs  about  7|  pounds,  and 
a  gallon  of  spirits  of  turpentine  about  7  pounds. 


336  APPENDIX. 

One  pound  of  prepared  white  lead  paint  will  cover  of  first  coat 
about  4  square  yards,  and  of  subsequent  coats  from  4^  to  5^  square 
yards. 

A  day's  work  for  a  painter  is,  of  plain  outside  work,  from  80  to  100 
square  j^ards,  and  of  inside  work,  from  40  to  65  square  yards. 

87.  Each  of  the  two  sides  of  the  roof  of  a  certain  building 
is  34  feet  long  and  25  feet  wide.  How  many  shingles  will 
be  required  for  it  if  laid  4  inches  to  weather,  and  how  many 
4-penny  nails  must  be  allowed  ?  What  will  be  the  cost  of 
laying  the  shingles,  at  $  2.50  per  day  for  labor  ? 

88.  How  many  clapboards,  laid  4  inches  to  the  weather,  will 
be  required  to  cover  the  side  pf  a  building  30  feet  high  and 
63  feet  long,  no  allowance  being  made  for  openings  ? 

89.  A  close  board  fence,  4  feet  high  and  64  feet  long,  is  to 
be  painted  both  sides  with  two  coats  of  white  lead  paint.  When 
the  lead,  ground  in  oil,  is  9  cents  a  pound,  and  linseed  oil  72 
cents  a  gallon,  what  will  the  paint  required  cost  ?  How  much 
must  be  paid  a  painter  for  putting  it  on,  if  he  covers  80  square 
yards  a  day  at  $2.50  ? 

90.  A  room  is  20  feet  long,  18  feet  wide,  and  10  feet  high. 
Allowing  108  square  feet  for  openings  and  mopboards,  how 
many  laths  will  be  required  for  its  ceiling  and  walls  ?  What 
will  be  the  cost  of  nails  for  setting  them  at  4^^  cents  a  pound  ? 

91.  A  hipped-roof  barn  is  64  feet  long,  40  feet  wide,  and 
20  feet  high  to  the  roof.  Allowing  360  square  feet  for  open- 
ings, how  much  will  rough  boards  for  the  sides  cost  at  $  20  a 
thousand  feet,  and  how  much  should  each  of  two  men,  at  $  2.25 
a  day,  be  paid  for  putting  on  the  boards  ? 

PBOGRESSIONS. 

539.  A  Series  of  numbers  is  a  succession  of  numbers,  in- 
creasing  or  decreasing  according  to  some  fixed  law. 

540.  The  Terms  of  a  series  are  the  numbers  forming  the 
series. 


APPENDIX.  337 

The  first  and  last  terms  are  called  the  extremes,  and  the  in- 
tervening terms  the  means.     Thus, 

3,  6,  9,  12,  is  a  series  in  which  3  and  12  are  the  extremes, 
and  6  and  9  the  means. 

541.    A  series  is  ascending  or  descending ,  according  as  the 
series  increases  or  decreases  from  the  first  term. 


ARITHMETICAL    PROGRESSION. 

542.    An  Arithmetical  Progression  is  a  series  of  numbers 
which  increase  or  decrease  by  a  common  difference.     Thus, 

2,  4,  6,  8,  10,  12,  is  an  ascending  series, 
12,  10,  8,  6,  4,  2,  is  a  descending  series, 

in  each  of  which  2  is  the  common  difference. 

To  find  either  Extreme. 

92.  The  first  term  of  an  arithmetical  progression  is  3,  and 
the  common  difference  2.     What  is  the  fifth  term  ? 

Solution. 

1st  term  =  3.  3d  term  =  3  +  (2  X  2). 

2d  term  =3  +  2.  4th  term  =  3  +  (2  X  3). 

5th  term  =  3  +  (2  X  4)  =  11. 

93.  The  fifth  term  of  an  arithmetical  progression  is  11,  and 
the  common  difference  2.     What  is  the  first  term  ? 

Solution. 

6th  term  =  11.  3d  term  =  11  —  (2  X  2). 

4th  term  =11—2.  2d  term  =  11  —  (2  X  3). 

1st  term  =  11  — -  (2  X  4)  =  3. 
Hence, 


338  APPENDIX. 

543.    To  find  an  extreme, 

Multiply  the  common  difference  by  the  number  of  terms  les$ 
one,  and  the  product  plus  the  smaller  extreme  will  be  the 
larger  ;  or,  the  larger  extreme  minus  the  product  will  he  the 
smaller, 

94.  The  number  of  terms  of  an  arithmetical  progression  is 
100,  the  common  difference  3,  and  the  first  term  5.  What  is 
the  last  term  ? 

95.  A  man  bought  34  yards  of  cloth,  and  agreed  to  give 
12  cents  for  the  first  yard,  Vl\  cents  for  the  second  yard,  12  § 
cents  for  the  third  yard,  and  so  on.  What  did  the  last  yard 
cost  him  ? 

96.  A  man  travels  10  days,  increasing  each  day's  travel  by 
I  of  a  mile.  If  he  goes  the  last  day  17  miles,  how  many  miles 
did  he  start  with  ? 

97.  If  16  persons  give  in  charity,  and  the  first  gives  5 
cents,  the  second  9  cents,  and  so  on  in  arithmetical  progres- 
sion, how  much  does  the  last  person  give  ? 

To  find  the  Sum  of  the  Terms. 

98.  The  first  term  of  an  arithmetical  progression  is  2,  the 
last  term  12,  and  the  number  of  terms  6.     What  is  the  sum  of 

all  the  terms  ? 

Solution. 

Let    2,        4,         6,        8,       10,      12,  be  an  arithmetical  series, 
and      12,      10,         8,        6,        4,        2,  be  the  series  reversed. 

14  -j-  14  _|-  14  _[-  14  -j-  14  _[_  14  =  84,  twice  the  sum  of  the 
series. 

But  84  =  (2  -}-  12)  X  6,  or  the  sum  of  the  extremes  multiplied 
by  6  ;  and  half  of  84,  or  (^  +  1^)  X  6  __  ^^^  ^^^^  ^^  ^^^  ^^■^\^^. 

Hence, 

544.    To  find  the  sum  of  the  terms, 

Multiply  the  sum  of  the  extremes  by  the  number  of  terms, 
and  take  half  the  product. 


APPENDIX.  339 

99.  The  first  term  of  a  series  is  2,  the  last  term  478,  and 
the  number  of  terms  S6,     What  is  the  sum  of  the  series  ? 

100.  A  man  agreed  to  labor  12  months.  For  the  first 
month  he  was  to  be  paid  $  7,  and  for  the  last  $  51.  If  he  was 
to  receive  the  same  addition  to  his  wages  each  successive 
month,  what  sum  would  he  receive  for  his  year's  labor  ? 

GEOMETRICAL  PROGRESSION. 

545.  A  Geometrical  Progression  is  a  series  of  numbers 
which  increase,  or  decrease,  by  a  common  rate  or  ratio.    Thus, 

3,  9,  27,  81,  243,  is  an  ascending  series  ; 
243,  81,  27,  9,  3,  is  a  descending  series. 
In  the  first  series  the  rate,  or  ratio,  is  3,  and  in  the  last  J. 

To  find  any  Term. 

101.  The  first  term  of  a  geometrical  progression  is  4,  the 
rate  2,  and  the  number  of  terms  5.     What  is  the  last  term  ? 

Solution. 

1st  term  =  4.  3d  term  =  4  X  2^. 

2d  term  =4X2.  4th  term  =  4  X  21 

5th  term  =  4  X  2*  =  64. 

102.  The  first  term  of  a  geometrical  progression  is  1458, 
and  the  rate  J.     What  is  the  seventh  term  ? 


Solution. 

1458 

729        "729" 
Hence, 


&  =  W9-' ''''><  i-. 


546.     To  find  any  term, 

Multiply  the  first  term  by  that  power  of  the  ratio  whose  ex- 
'ponent  is  equal  to  the  number  of  terms  less  one. 

103.    The  first  term  of  a  series  is  10,  the  rate  20,  and  the 
number  of  terms  5.     What  is  the  last  term  ? 


340  APPENDIX. 

-     104.   When  the  first  term  is  $  120,  the  ratio  1.06,  and  the 
number  of  terms  4,  what  is  the  last  term  ? 

105.  What  will  $  50  amount  to  in  4  years  at  6  %  compound 
interest  ? 

To  find  the  Sum  of  the  Series. 

106.  A  geometrical   progression   consists   of  2,  6,  18,  54, 
the  ratio  being  3.     What  is  the  sum  of  all  the  terms  ? 

Solution. 

6  +  18  +  54  +  162  =  3  times  the  series. 

2  +  6  +  18-1-54  =  once  the  series. 

162  —  2  =2  times  the  series. 

162  —  2  en       ^i, 

•  =  80  =  the  series. 

2 

Subtracting  like  terms  of  once  the  series  from  three  times  the 

series,  there  remains  162  —  2,  as  two  times  the  series,  or  80  as  the 

sum  of  the  series. 

Hence, 

547.    To  find  the  sum  of  the  series, 

Multiply  the  last  term  by  the  ratio,  subtract  the  first  term, 
and  the  remainder  divided  by  the  ratio  less  one  will  give  the 


107.  What  is  the  sum  of  a  geometrical  progression  whose 
extremes  are  6  and  768,  and  ratio  2  ? 

108.  The  first  term  of  a  geometrical  progression  is  10,  the 
ratio  J,  and  the  number  of  terms  5.  What  is  the  sum  of  the 
series  to  the  nearest  hundredth  ? 

109.  The  first  term  of  a  geometrical  series  is  $  100,  the 
rate  1.06,  and  the  number  of  terms  4.  What  is  the  sum  of 
the  series  ? 

110.  A  lady,  wishing  to  purchase  10  yards  of  velvet,  thought 
$4  a  yard  too  high  a  price.  She,  however,  agreed  to  give  1 
cent  for  the  first  yard,  4  cents  for  the  second,  16  cents  for  the 
third,  and  so  on.     What  was  the  cost  of  the  velvet  ? 


APPENDIX.  341 

COLLEGE  ENTRANCE-EXAMINATION 
PAPERS. 

548.    Brown  University, 

1.  Divide  fifty  millionths  by  six  hundred  twenty-five  ten- 
thousandths,  and  express  the  quotient  in  words. 

2.  A  merchant  owned  \\  of  a  stock  of  goods  ;  |  of  the  whole 
stock  were  destroyed  by  fire,  and  -^^  ^^  ^^^  remainder  damaged 
by  water.  How  much  did  the  merchant  lose,  provided  the 
uninjured  goods  were  sold  at  cost  for  $  4200,  and  the  damaged 
at  half  the  cost  ? 

3.  How  many  hektol iters  of  oats  can  be  put  into  a  bin  that 
is  2"^  long,  1.3'"  wide,  and  1.5'"  deep  ? 

4.  Sold  a  village  lot  for  1 230,  which  was  8  per  cent  less  than 
cost.  Had  it  been  sold  for  $  300,  what  would  have  been  the 
gain  per  cent  ? 

549.    Dartmouth  College, 

2f  -^  -^  X  2 

1.  Y =  ^ 

2-1-5 

2.  Find  the  sum  and  product  of  J,  J,  f . 

3.  Find  the  cube  root  of  3845672000. 

4.  Find  the  square  root  of  3534400.5. 

5.  A  platform  bears  a  weight  of  100  lb.  per  square  foot. 
What  is  the  weight  in  kilograms  per  square  meter  ? 

6.  A  horse  that  cost  6 J-  per  cent  of  %  25000  was  sold  for 
%  1000.     What  was  the  loss  per  cent  ? 

550.    Trinity  College, 

1.  Subtract  thirty  million  twenty-six  thousand  three  from 
45007021.  Find  what  number  must  be  added  to  the  difference 
to  make  one  hundred  million,  and  write  the  answer  in  words. 


342  APPENDIX. 

2.  The  sum  of  f  and  ^  is  diminished  by  ■^.  How  many 
times  does  the  difference  contain  f^j  of  the  sum  of  J,  ^,  and  ^  ? 

3.  Divide  375  by  .75,  and  .75  by  375,  and  find  the  sum  and 
the  difference  of  the  quotients. 

4.  A  loaded  wagon  weighs  2  T.  3  cwt.  48  lb. ;  the  wagon  it- 
self weighs  18  cwt.  75  lb.  The  load  consists  of  215  packages, 
each  of  the  same  weight.  Find  the  weight  of  each,  and  re- 
duce it  to  grams  and  kilograms. 

5.  Define  interest,  and  give  and  explain  the  rule  for  com- 
puting the  interest  on  any  sum  of  money  for  any  time  and  at 
any  rate  per  cent. 

6.  Extract  the  square  root  of  184.2  to  three  places  of  deci- 
mals. 

551.    Harvard  University. 
i.   Find  the  greatest  common  divisor  of  315,  504,  and  441. 

2.  Express  as  a  decimal  f  X  ^  ,, ,  ^    o  ^     • 

3.  How  many  hektoliters  are  there  in  57  gallons  3 J  pints  ? 

1  liter  z=  0.2642  gal. ;  1  gallon  =  8  pt. 

4.  How  much  paper,  1\  yd.  wide,  will  be  needed  to  paper 
the  walls  of  a  room  10  feet  high,  18  feet  long,  12  feet  broad? 

5.  I  sold  a  lot  of  sugar  for  $  230,  and  thereby  lost  8  per 
cent  of  the  cost.  What  per  cent  should  I  have  gained  if  I  had 
sold  the  sugar  for  $  300  ? 

552.     Yale  College. 

1.  Find  the  value  of  (  5-?-  of  —  )  divided  by  ^,  and  exv 

tract  the  square  root  of  the  quotient  to  two  decimal  places. 

2.  Find  a  fourth  proportional  to  .37,  8.9,  4.3,  and  extract 
the  cube  root  of  it  to  two  decimal  places. 


APPENDIX.  343 

3.  Eeduce  16  rods  2  feet  and  8  inches  to  the  decimal  of  a 
mile. 

4.  What  is  the  length  in  meters  and  decimeters  of  a  side  of 
a  square  which  contains  .1335  are  ? 

553.  Dartmouth  College, 
^   41  +  2^  -^  I  ^  , 

'  61  -  If  X  t      • 

2.  Find  the  least  common  multiple  and  the  greatest  com- 
mon divisor  of  6,  8,  20,  and  36. 

3.  How  many  meters  in  25  feet  ? 

4.  Find  the  square  root  of  3530641. 

5.  Gold  was  quoted  at  $1.12|.  What  was  a  one-dollar 
greenback  worth  ? 

6.  1 1200  includes  a  sum  to  be  invested,  and  a  commission 
of  five  per  cent  of  the  sum  to  be  invested.  What  is  the  sum 
to  be  invested  ? 

554.  Cornell  University. 

1.  Define  a  composite  number,  a  factor,  an  abstract  number, 
the  cube  root  of  a  number,  equation  of  payments. 

2.  What  is  the  value  of  50  lb.  8  oz.  of  gold  at  1 20.59J  per 
ounce  ? 

3.  Given  the  meter  equal  to  39.37  inches,  reduce  one  mile 
to  kilometers.     Give  the  metric  table  of  weights. 

4.  Divide  f  of  7f  by  f  of  12^^J.  Prove  the  result  by  re- 
ducing the  fractions  to  decimals,  and  working  the  example 
anew. 

5.  A  man  said,  "  I  will  spend  half  my  income,  save  a  third 
of  it,  and  devpte  a  fourth  to  business.'^  His  income  was  $  780 
a  year.  Point  out  his  blunder,  and  divide  his  income  rightly 
in  the  proportion  intended  by  him. 

6.  How  long  must  $  125  be  on  interest  at  7J  per  cent  to 
gain  $  15  ? 


344  APPENDIX. 

7.  Eeceived  6  per  cent  dividend  on  stock  bought  at  25  per 
cent  below  par.  What  rate  of  interest  did  the  investment 
pay? 

a   Find  the  cube  root  of  .726572699. 

555.     Yale  College, 
1.   Eeduce  to  a  common  denominator,  and  add: 


.3  9 

'  15'  4'  ^"""^  10- 


/3      4      5\   ^ 
\^^  5^  9/  11 

2.  Divide  (1-1)  by  1. 

3.  Find,  to  three  decimal  places,  the  value  of  -— 3  • 

V3 

4.  Find  the  fourth  term  of  a  proportion  of  which  the  first, 
second,  and  third  terms  are,  respectively,  3.81,  0.056,  1.67. 

5.  Eeduce   3  K   13  sq.  rd.   8  sq.  ft.   to  the  decimal  of   an 
acre. 

6.  {a)  In  a  board  4 ""  long  and  0.4  ""  wide,  how  many  square 
decimeters  ? 

{h)  Divide  2700  "'by  90  ^^ 

556.    Harvard  University. 


1.   Simplify 


1  +  3  + 


(S) 


t)-^ 


2.  Find  the  length  in  dekameters  of  the  side  of  a  square, 
the  area  of  which  equals  the  area  of  a  rectangle  which  is  1  kilo- 
meter 8  meters  long,  and  4jf  hektometers  wide. 

3.  Find  the  least  common  multiple  of  the  even  numbers 
from  10  to  20  inclusive. 


APPENDIX.  345 

4.  The  interest  on  $2500  for  2  months  12  days  is  $45. 
Find  the  rate. 

5.  If  25  men,  working  8  hours  a  day,  do  |  of  a  piece  of  work 
in  24  days,  in  how  many  days  of  10  hours  each  will  30  men 
finish  the  piece  of  work  ? 

557.     University  of  the  State  of  New  York. 

1.  Copy  and  add :  20570,  6206,  98.007,  63000,  426.000626, 
4287,  63.961, 102030,  405.0607,  8090,  543.21,  1028848.414995. 

2.  Express  by  Arabic  notation,  MDXCYDCCCLXIV. 

3.  Express  by  Roman  notation,  84796. 

4.  Numerate  20567189.004321098. 

5.  Divide  31984875832  by  96813. 

6.  Find  the  value  of 

(28  -  7)  X  6  +  (92  4-  7)  -^  9  -  (86  +  10)  -^  12. 

7.  Divide,  using  cancellation, 

15  X  80  X  27  X  28  by  7  X  20  X  8. 

8.  Change  /y,  Jf ,  t^%,  and  J,  to  similar  fractions  having 
their  least  common  denominator,  and 

9.  Reduce  their  sum  to  decimal  form. 

10.  Find  the  greatest  common  divisor  of  7955,  8769,  6401. 

11.  How  much  must  be  paid  for  making  52  rd.  14  ft.  8  in. 
of  fence,  at  $  0.75  per  foot  ? 

12.  A  traveler,  on  reaching  a  certain  place,  found  that  his 
watch,  which  kept  correct  time  for  the  place  he  left,  was  2  h. 
22  min.  slower  than  the  local  time.  Had  he  traveled  east- 
ward or  westward,  and  how  far,  in  circular  measure,  had  he 
come  ? 

13.  What  per  cent  (expressed  in  words)  of  30000  bushels 
are  50  bushels  ? 

14.  What  number  diminished  by  36%  of  itself  =  336  ? 


346  APPENDIX. 

15.  What  is  the  value  of  a  lot  70  rd.  long  and  20  rd.  wide, 
at  $  47.25  per  acre  ? 

16.  A  cistern  has  3  pipes.  The  first  will  fill  it  in  12  hours, 
the  second  in  16,  and  the  third  in  18  hours.  If  all  run  to- 
gether, in  what  time  will  they  fill  it  ?  (State  this  example  as 
a  proportion,  if  you  can.) 

17.  What  is  the  difference  between  the  simple  interest  on 
$328  for  2  y.  7  mo.  at  7%,  and  the  compound  interest  on  the 
same  sum  for  the  same  time  at  6  %  ? 

18.  Find  the  balance  due  March  4,  1881,  on  a  note  dated 
Jan.  1,  1879,  for  $  580,  at  5  %,  on  which  a  payment  of  I  85  has 
been  made  every  6  months,  using  the  United  States  rule. 

19.  How  much  should  be  discounted  on  a  bill  of  $3725.87, 
due  in  8  mo.  10  d.,  if  paid  immediately,  money  being  worth 
5%? 

20.  Bought  bonds  at  115,  and  sold  at  110,  losing  $  300. 
How  many  bonds  of  $  1000  each  did  I  buy  ? 

21.  If  A  puts  in  $  4000  capital  for  8  months,  B  $  6000  for 
7  months,  and  C  $  3500  for  1  year,  and  they  gain  $  2320, 
what  is  each  partner's  share  of  the  gain  ? 

22.  If  5  horses  eat  as  much  as  6  oxen,  and  8  horses  and  12 
oxen  eat  12  tons  of  hay  in  40  days,  how  much  hay  will  7  horses 
and  15  oxen  eat  in  65  days  ? 

23.  Find  the  value  of  V'0.000238328. 

24.  A  steamer  goes  due  north  at  the  rate  of  15  miles  an 
hour,  and  another  due  west  18  miles  an  hour.  How  far  apart 
will  they  be  in  6  hours  ? 

25.  Find  the  cost,  at  30  cents  per  sq.  yd.,  of  cementing  the 
bottom  and  sides  of  a  cubical  cistern  that  will  hold  300  barrels. 

26.  What  is  the  area  of  a  circle  5  ft.  in  diameter  ? 

27.  What  is  the  difference  between  5  square  feet  and  5  feet 
square  ?    Illustrate  by  a  diagram. 


ANSWERS. 


Art.  87. 

44.  1009 

45.  669. 
J^6.  698. 
47.  759. 
4S.  $989. 
Ji9.  J499. 
50.  $1465. 
79.  1484. 
^(?.  1149. 

81.  $31,355. 

82.  $39,928. 

83.  $61,665. 
^4.  $317.40. 


Art.  39. 

85.  689. 

86.  1978. 

87.  2396. 
^5.  15485. 
^P.  2052. 
90.  9788. 
9i.  2018.7. 
92.  143.91 
P5.  $131.31. 

94.  $100.66. 

95.  $393.30. 

96.  $230.05. 
.97.   3018. 
9^.  3443. 
99.  7736. 

100.  2023. 
i9i.  2026. 
10^.   16986. 


i^^. 

176.40. 

Art.  46. 

i94. 

153.89. 

4S.  332. 

295. 

281.72. 

4^.  223. 

106. 

5233.97. 

47.  205. 

107. 

$125.65. 

48.  222. 

108. 

$511.69. 

49.  1114 

109. 

$168.08. 

50.  3212. 

110. 

$532.40. 

51.  2213. 

111. 

998. 

52.  1118. 

112. 

4391. 

53.  1221. 

113. 

$9665.68.^: 

54.  $325. 

in. 

5619. 

76.  309. 

115. 

$75.13. 

77.  192. 

116. 

318381. 

78.  192. 

117. 

$145.17. 

79.  46.65. 

118. 

$360. 

80.  803.153. 

119. 

$14170.70. 

81.  $391.05. 

120. 

2815. 

82.  $55.14. 

121. 

24445. 

83.  $338.80. 

122. 

313.54. 

123. 

150.390. 

Art.  48. 

124. 

29002  ft. 

84.  5196. 

125. 

3578^92. 

85.  4969. 

126. 

$1046.87. 

86.  1859. 

427. 

$175i30 

87.  1056. 

128. 

62611. 

88.  29962. 

129. 

$39320. 

89.  3541. 

130. 

1323925.63. 

90.  56.39. 

131. 

1863189. 

9i.  14.251. 

132. 

563972744718. 
509006545503.418. 

92.  $6.27. 

133. 

95.  $83.96. 

134. 

323497. 

94.  $95.81. 

135. 

340522022. 

95.  $29.99. 

136. 

1380855.262. 

97.  34456. 

137. 

$32545.24. 

98.  97820. 

138. 

$24005.79. 

99.  22968. 

us 


ANSWEES. 


100.  9903. 

101.  9154. 

102.  1. 

103.  6.552. 

104.  811.95. 

105.  9615.5 
i{?^,  78.44. 

107.  $486.57. 

108.  $1836.75. 

109.  5541. 

110.  26983. 
iii.  11001. 
112.   107.91. 
ii,5.  389. 
114,   $740.75.' 
il5.  163864. 
ii6.  $4066.94. 
117.   269535. 
ii,5.  267369. 
119.   1785837. 


i^a  $291.25. 

i^i.  $3527.82. 

122.  891;  460;  280. 

123.  $173.32. 

124.  1382. 
i^5.  1330  miles. 

126.  $5555.75. 

127.  380247. 

128.  $1031.32. 

Art.  68. 

38.  3024. 

S9.  13701. 

^(?.  5545. 

41.  49608. 

^.  20496. 

4^.  5216. 

.^4.  86415. 

.^.  218709. 

47.  2S0.56. 

i?.  30.675. 

4.^  3271.8. 

50.  49899.71. 

5i.  $497.40. 

52.  $1155.282. 

53.  $6588.00. 

54.  $76838.70. 
81.  2S2;U. 


82.  60525. 

83.  10150. 
<?.^.  55114. 
85.  115842. 
5^.  501030. 
<57.  10003.121. 
88.  18102.25. 


Art.  60. 


89. 

90. 

91. 

92. 

93. 

94. 

95. 

96. 

97. 

98. 

99. 
100. 
101. 
102. 
103. 
104. 
105. 
106. 
107. 
108. 


266843. 

200718. 

7684. 

$2923.83. 

31055.6. 

4824.342. 

9309.65. 

4378905. 

$159438.30. 

41105.24. 

2951129.6. 

$46471.68. 

8210.90. 

66909.44. 

27835.83. 

$292636.36. 

1054.875. 

59073353. 

14518308. 

$675729.95. 

109.  4193735. 

110.  26502.93. 
325631.74. 
53794576. 
$11545.45. 
40444770. 
299023.58. 
68697853. 
$68479.40. 

118.  $86214.193. 

119.  48085. 

127716. 

4143204. 

50684,13. 

58501.12. 

$2062.50. 

125.  205732.57. 

126.  639446.5. 
m.   8S68G4  86. 
128.   349732  16 


111. 
112. 
113. 
114 
115. 
116. 
117. 


121. 

122. 
123. 
124. 


129.  $5642.455. 

130.  $9810.944. 

131.  $219187.65. 

132.  264798  bush, 

133.  3575806.65. 

135.  473970. 

136.  7854000; 138000 

Art.  61. 


138. 

814000. 

139. 

274470. 

140. 

493020. 

141. 

603900. 

142. 

$67244.10. 

143. 

2760720. 

144 

$  3208100. 

145. 

2051636.8. 

146. 

27306000. 

147. 

$  23364000. 

14s. 

16320000  miles. 

149. 

4992000  acres. 

150. 

$110.75. 

151. 

99^4. 

152. 

$1061.40;  could 

havesaved$1129.00 

153. 

732  miles. 

154. 

1632001b. 

155. 

45981b. 

156. 

$86.45. 

157. 

150696. 

158. 

$1353.50. 

159. 

$100. 

160. 

$268.95. 

161. 

$  4409. 

162 

$1326. 

Art.  71. 

36. 

383. 

37. 

821. 

38. 

971. 

39. 

1142. 

40 

58.6. 

41- 

12.23. 

^. 

11.90. 

4^. 

4.467, 

4^. 

715i 

J^6.  678^. 
47.  612|. 
4S.  84234. 
4^.  6731-. 

50.  13.34. 

51.  60.71J. 
5^.  4.348f. 
5^.  405. 

54.  205f  miles. 

55.  $5.45. 

56.  23131 

Art.  72. 

^9.  470g\. 

70.  234. 

7i.  1359f4. 

72.  5.63. 

7^.  1.063. 

7.4.  50.5. 

Art.  74. 

76.  $3.18. 

77.  $32.02. 
7<^.  990ff2, 
75.  10.07^^. 
SO.  965A%. 
^2.  4.47fH- 
82,  5.509^m. 
5^.  $8.503jX. 
<5^.  505f|f|. 
^5.  1487Hf- 
56.  75|}f. 

87.  2.056^^. 

6',^.  8.75f|f 

<59.  1208fff. 

50.  245 3XV 

5i.  6888i|. 

92.  $125. 

.95.  24. 

5^.  $.6848.7641. 

95.  26iV_. 

•96.  3.44fff. 

.97.  3007. 

98.  114^%. 

59.  342. 
WO.  $52.88. 
20i.  $136f||. 
103.  61.87ff. 


ANSWERS. 

iO.4.  106.558i|. 

105.  $31.25. 

i06.  137.125. 

i6>7.  30303gV. 

Art.  76. 

110.  $19.65. 

111.  39.62;  3.962. 
ii^.  45.54. 

113.  137ff. 

ii^.  1090.18iW. 

^^^-  mm- 

116.  494ii|-|^sec. 

117.  1024||||.' 

118.  768f4|f. 

119.  5888ij^%V 

i;^o.  11956mi 

121.  1.542Aff 
i^^.  .529fU. 
123.  977iMf. 
^^^.  1.788i||. 
i^5.  585,W^. 

126.  im^s%\. 

127.  100|f|. 

128.  1016400. 

129.  30515^^. 

i^o.  $l.ll|ii 

131.  3^^. 
i^-^.  .871|^. 


349 


138.  2555.         , 

135    3.  '^ 

i<^6.  |26.77ftV 

137.   14. 

i^.?    640.    > 

i55.  106. 

7.^5.  64. 

I4I'  35;  and  1400  1 

mi.  over, 
i^.  750. 
i4^.  108. 
lU   71. 
i-^.  217. 

Art.  76. 

.?5.  3207. 
^.f  39817. 


35. 

36. 


40. 

41. 

42. 

4S. 
44- 
45. 

46. 

47. 

48. 

4^. 

50. 

51. 

52. 

53. 

54. 

55.^ 

56. 

57. 

58. 

59. 

60. 

61. 

62. 

63. 

64 

65. 

66. 

67. 


1492. 

$  2063.38. 

12616. 

$371.14. 

$  15.30. 

6. 

$35405. 

28618. 

$0.14. 

$0.24fio 

$4.71^11. 

27. 

$8. 

$1820.85. 

873. 

$3632.25. 

137ff  hhds. 

$123. 

192. 

1808. 

$7137. 

17» 

759. 

550. 

17. 

$  269880. 

1160. 

13. 

$5817. 

$4.53f. 

215. 

$4064,  gain. 


Art.  82. 

8.  22,  3,  7;    2^3-^ 

2S,  5. 

9.  2,  3,   7,  11;    2«, 

32;  2^  32,  7. 

Art.  83. 

10.  2,  3,  5,  7. 

11.  22,  32,  7,  11. 

12.  2,  3,  71. 
18.  3,  5,  11,  37. 
14   3,  5,  7,  11. 
15.  24,  52,  7. 


350 


ANSWERS. 


16.  22,  32,  5,  19. 

17.  23,  3,  52,  13. 

18.  22,  4007. 

19.  38,  72,  13. 
^0.  32,  31,  37. 
21.  2^  11,  71. 
^2.  37. 

^«5.  3,  7,  11,  19. 

;^^.  22,  32,  52,  7. 

;^5.  7,  11,  17,  31. 


Art.  86. 

35.  12^. 

36.  3|. 
57.  109. 
5<?.  24. 
5P.  9J. 
-^0.  7. 
41-  19.    . 
4^.  8. 
J,S.  4. 

-^.  $0.30. 

Art.  90. 

50.  9. 
5i.  6. 

Art.  91. 

52.  45. 
5<^.  15. 

54.  3. 

55.  12. 

56.  2. 

57.  16. 
5^.  9. 
69.  3. 

6?d?.  3  rt. 

Art.  92. 

5;?.  48. 

63.  25. 

64.  22. 


65.  86. 

66.  55. 

67.  39. 

68.  2. 
6P.  87. 
76>.  56. 
71.  28. 
7^.  6. 
73.  8. 
7.4.  12. 

75.  31. 

76.  14. 

77.  696. 

Art.  96. 

83.  210. 
^4.  1260. 

Art.  97. 

85.  1848. 

86.  504. 
^7.  1320. 
^^.  2835. 

89.  72. 

90.  86100. 

91.  7000. 
9^.  7560. 
P«5.  7200. 
9J^.  7920. 
95.  350. 


96.  23,  32,  5,  7. 

97.  27,  32. 
9<?.  13. 
99.  492. 

iOO.  80. 

iOi.  12. 

102.  1777. 

i65  54528. 

104.  6. 

i65.  105. 

Art.  112. 

35.  If 

37. 
38. 


39.  m 

41-  If. 


Art.  114. 


54. 


Art.  US. 


6^.  |. 
65.  Iff 

66    ai. 


Art.  116. 


85.  i||^. 
^6    IjV-. 

87.  ^^. 

88.  4^. 
^^-  fit. 


^{1^. 


97.  l{| 
i05.  43. 
106.  63>f. 


ANSWERS. 


351 


Art.  117. 

107.  9. 

108.  27f 

109.  169iV 

110.  1. 

Ill-  1H|. 

112.  l^V 

113.  7H- 
li^.  128. 
115.  llOff. 

118.  90|f. 

119.  91|f. 

120.  100  iW. 
1^1.  95fff 


Art.  120. 


isi-  il 

133.  ^^, 


to"' tA* 


Art.  121. 

135. 
136. 

138. 

139.  ii 

m-  fA.  1%,  T^v 

6  6 

14(10 

J^^      M      36     M. 

Art.  123. 

156.  2H. 

Art.  124. 

158.  If. 

159.  2^\. 

160.  !«. 


lei.  2ff 

165.  8f|. 

16^.  2|||. 

165.  2|4|. 

166.  9^1^. 

Art.  125. 

167.  40|. 
16^.  488^. 
169.  27i. 
17(?.  313|f. 

Art.  126. 

182. 
183. 

m.  ^. 

Art.  127. 

185. 
186. 
187. 

188.  Uf 

189.  ^. 

190.  \. 

191.  T*k. 
19^.  X. 
193.  -,y 

m.  eV- 

i^5-  17ff. 

196.  21f|. 

197.  SX. 

198.  6^. 

199.  13^. 
^9(?.  280|f. 
201.  4^. 
^(?^.  ^5^. 
203.  2ff . 

Art.  128. 

212.  1|. 
;^15.  154 
214.  37| 
^15.  49. 
;^16.  lOH- 
217.  2A\. 


;^1^.  43^. 

219.  144. 

^^1.  1090^. 

^^^.  5131. 

223.  2927f 

^^.^.  10025. 

225.  $621. 

Art.  120. 

^5^.  84. 
233.   241}. 
^5-4.  325. 
235.   97f. 
^.?6.  88. 
;^,S7.  165iV. 
238.   17A. 
^59.  243A. 
241.   10724. 
;^4^.  1119|. 
^.4<^.  791. 
2U'   1719f. 
^-^.  1281J. 

Art,  131. 

257.   134. 

259.   l| 
^66. 
;^61.  2f. 


■5.  2| 
;^64.  f 
265.  Jt589A' 
^66.  82f^. 
^67.  Iff 
;^6^.  25^. 
^69.  $122||. 
270.  U^. 

Art.  132. 

285.  ^4^. 

;^^7.    ■ 

\288. 


352 


ANSWERS. 


^89. 
'^91. 


A 


^92. 

15fJ. 

293. 

3H. 

29 J,.  «6f. 

SOJf,. 

mi 

305. 
306. 

29A. 

874. 

52|. 

307. 

308. 

170. 

309. 

151iV 

310. 

40. 

311. 

162. 

812. 

ml 

313. 

su. 

64. 

815. 

16. 

316. 

7. 

827. 

141. 

828. 

iV 

Art.  133. 

829. 

|. 

830. 

^. 

S31. 

WA. 

832. 

lA- 

833. 

1 

334.  AV 

S35. 

2tV 

336. 

HH- 

837. 

2||. 

838. 

iilj. 

339. 

^M- 

&40. 

u^. 

341. 

41  hours. 

34s. 

ill- 

344. 

m. 

345. 

16|f. 

34B.  i." 

347. 

^TS' 

848. 

-  |. 

349. 

■r^. 

350. 

45  days. 

851. 

4  hours. 

Art.  134, 

S59 

f 

360 

4. 

367.  {%. 

368.  if. 

^ra  112. 

.^7i.  $52.50. 

372.  4|. 

^7^*.  |'4iVo- 


Art.  136. 

,^^9.  1672. 
390.  1863. 
^9i.  6400. 
«^9^.  520. 
393.  11400. 
<?94.  330. 
395.  |. 
^96.  25^. 


«?9<5.  $6550. 
399.  $2625. 
4(?a  $8919. 


401.  52^. 

40^.  $f. 

40^.  llf 

404.  $6.40. 

^(?5.  103  J. 

4O6.  2|f. 

^7.  73|  mm. 

4O8.  «13^. 

4^9.  A. 

-jia  7p 

4i^.  1  to  1 

4i<?.  48. 

414.  U. 

425.  l|. 

4I6.  $30. 

4/7.  8,1^. 


pieces  and  2  ft. 


4^^.  $414. 
419.  $561. 

Art.  136. 

«^5.  576. 
^6.  15552. 
38.  l^Vo- 
^9.  18}f. 
4(?.  15^. 
41.  34. 

4^.     2,;V0- 

44.     27. 

^.   sVi- 
4(5.   76. 

47.  2j|.    • 

-^.  « . 

49.  13^V 

50.  '^ 
51. 

52.  ^-^ 

53.  560. 

54.  \h 

55.  $141. 

56.  $107U. 

57.  $1^. 

58.  19f. 

59.  $5.68;  $59.64, 

60.  70  cents. 
(5i.  56. 

62.  llf 
6^.  If 

(A's$300. 

64.  ^B's$569. 
(C's$100. 

65.  ij. 

66.  $1485. 

67.  31J. 
6^.  $6f|. 
69.  $f. 
7(?.  $500. 

^.  C  A  560  gal. 
^^'  iB  400  gal. 
72.  2/^^. 

Art.  142. 

54'  71.5. 


ANSWERS. 


353 


55. 


56. 


19.000. 
I  43.600. 
I    0.640. 

53.000. 

15.60. 
4.70. 
(13.00. 
(  18.0156. 
57.  I  401.6000. 
( 176.4700. 

63.  A. 

64.  \. 


Art.  143. 


65.  ^V- 


84.  .6. 

86,  .416f. 

87.  .3636^\. 

Art.  144. 

?5.  .275. 

8C.  .4375. 

90.  .03125. 

91.  .7. 

92.  .32. 

93.  4.096. 

94.  4.015625. 

95.  2.09375. 

96.  5.0078125. 

97.  .0769^%. 

98.  .5833. 
£>P.  .9474. 

100.  .0933J. 

i(?i.  .135135. 


iO^.  1.232. 
103.  51.9383. 
i(?.^.  11.933877. 

Art.  145. 

111.  204.655. 

112.  132.912. 

Art.  146. 

115.  253.80. 

116.  9650. 
ii7.  0.014582. 
im  84.5688. 
119.  21.723. 
i^O.  26.461. 

121.  143.3282 

122.  0.00038665. 

123.  .0000505. 

124.  .0001292. 

125.  1.000000. 

126.  $160,875. 

127.  $61.7925. 

128.  0.01020201. 
139.  0.023. 

Art.  147. 

IJ,^.  125.36.^ 

U3.  0.756. 

lU.  10.01. 

11^5.  790. 

146.  0.081. 

147.  0.893+ 
i4^.  0.001365. 

4000.   .^ 

150000. 

1.12. 

360.984. 

15.004-f-. 

210. 


11,9. 
150. 
151. 
152. 
153. 
154. 


155. 

157. 

158. 
159. 


5  0.0403. 
I  400.0003. 
'  0.0087. 

0.00162. 

0.00Q009. 
5.325. 
8.9999.7. 


161. 

162. 
163. 
I64. 


167, 


160.  0.855. 

(  6345.3654. 
{  4.1958. 
(0.0000495. 
500. 

A- 

23.9268  sum. 

165.  14^54368  days. 

166.  612. 
$  2999. 45J  A's. 
$2249.62jB's. 
$  2249. 62J  C's. 

18817.92728. 

58.872. 

$  16444.9602. 

4.9835.  ._.. 

96. 

165.135.-- 

$0.18.— 

$447.70. 

$33|fi. 

1. 


168. 
169. 
170. 
171. 
172. 
173. 
174. 
175. 
176. 
177. 
178. 

179.  $219.54t\. 

180.  $5941.06. 

181.  7e>j\\. 

182.  68|||tons. 

183.  420. 

184.  4000. 
1^5.  3f. 

Art.  154, 

28.  $27.26. 

24.  $40.93. 

25.  $5143.15. 

26.  $19.36. 

27.  250. 

28.  $41.19. 
$94.86+. 
$48. 
44ff. 
760. 

$  163.12f 
50.95^^. 

156, 


J 


29. 
30. 
31. 


34 


Art, 

44.  $198. 

45.  $56. 

46.  $232.50. 


554 


ANSWERS. 


^7. 

98. 

JiS. 

18. 

49. 

^49. 

50. 

$679.50. 

51. 

2117. 

53. 

$5.10. 

54. 

$81.25. 

56. 

$248.60. 

57. 

$  904.042. 

59. 

$12. 

60. 

$12.50. 

61. 

$1.50. 

Art.  159. 

62. 

$90.47. 

63. 

$222.71. 

6J^. 

$58.55. 

65. 

$56.23. 

66. 

$104.78. 

67. 

$  226.50. 

68. 

$1156.25. 

69. 

$84. 

70. 

$443.35. 

71. 

$126.64. 

Art.  187. 

13. 

1279  pt. 

u. 

1311  cu.  ft. 

15. 

35790  lb. 

Art.  188. 

16.  562068  sq.  ft. 

17.  273749  cu.  in. 

18.  487  pt. 

19.  879  pt. 

20.  248160  ft. 

21.  419887  gr. 

22.  87320  lb. 

23.  31556930  sec. 
2^.  391256  cu.  in. 

25.  163734  sec. 

26.  3200  cu.  ft. 

27.  1306  ?i. 

28.  684592  min. 

29.  4320  sheets. 

30.  $3267. 
SI.  $6352.50. 


3S.  88  yd. 


35. 
86. 

q>>y 


is  sec. 


39.  48  lb. 

J^0.  326.7  sq.  ft. 

Ji.1.  568.08  min. 

Ji2.  6601.76  sq.  yd. 

Art.  189. 

51.  19  bu.  3  pk.  7  qt. 
1  pt. 

52.  48cu.yd.l5cu.ft. 

53.  17T.17cwt.901b. 


Art.  190. 


rd. 


5Jt.   12  A.  144  sq, 
144  sq.  ft. 

55.  5  cu.  yd.  23  cu. 
ft.  725  cu.  in. 

56.  60  gal.  3  qt.  1  pt. 

57.  13  bu.  2pk.  7qt. 
Ipt. 

58.  4^7  xrl. 

59.  72  lb.  10  oz.  15 
pwt.  7  gr. 

60.  43T.13cwt.201b. 

61.  365  d.  5  h.  48  m. 
50  sec. 

62.  8  cu.  yd.  10  cu.  ft. 
728  cu.  in. 

63.  45^  28'  54''. 

64.  25  cd. 

65.  40  gal.  3  qt.  0  pt. 
2gi. 

66.  67  wk.  6  d.  9  h. 
52  min. 

67.  4  bundles  1  ream. 

68.  1  A.  80  sq.  rd. 

69.  3  mi.  195  rd. 

mi. 

lb. 

r^  day. 

gal. 

■  bu. 

).'024  T. 

0.0075  A. 


79.  0.3945  day. 

80.  1.364  A. 

Art.  191. 

88.   71  rd.  1  ft.  10  in. 

90.  2  yd.  2  ft.  8.94  m. 

Art.  192. 

91.  68  sq.  rd.  155  sq. 
ft.  82f  sq.  in. 

92.  10  oz.  13  pwt. 
8gr. 

93.  232d.  6h.  32min. 
43  j\  sec. 

94.  248  rd.  4  yd.  2  ft. 
8  in. 

95.  671b.  4oz. 

96.  5  cwt.  64  lb. 

97.  4yd.  2  ft.  5.25  in. 

98.  18  h.  15  m.  50.4s. 

107.  f  mi. 

108.  U.53rd. 


Art.  193. 

110. 

AWoA. 

111. 

1  lb. 

112. 

fiiffy. 

9^  ini. 

113. 

114. 

0.6725  ctl. 

115. 

0.282  T. 

116. 

0.875  rd. 

117. 

0.761  d. 

Art.  194. 

118. 

;.. 

119. 

Im- 

120. 

tVi- 

121. 

O.J  25. 

122. 

11.17H- 

Art.  195. 

126.  63  cu.yd.  11  cu. 
ft.  842  cu.  in. 

127.  199  gal.  1  qt. 
12^.  77  d.    8I1.  26  m. 

56  sec. 
129.   64^  28'  32". 


ANSWERS. 


X 


131.  IT.  15cwt.  531b. 
182.   1()7  mi.  240  rd. 
133.  2  A.  49f^sq.  rd. 

135.  13  bu.  2  pk.  6  qt. 

136.  14  mi.      231  rd. 
5  yd.  0  ft.  2  in. 

137.  28  A.  76  sq.  rd. 

138.  16°  30'  51". 
189.  5oz.5pwt.20.8gr. 

140.  12  gal.  2  qt.  0  pt. 

141.  133  d.     4h.     39 
mill.  21.6  sec. 

14^.  ISlrd.  2Yd.  2  ft. 
8.72  in. 

Art.  196. 

lU-  5  y.  9  mo.  24  d. 

145.  1  y.  7  mo.  20  d. 

146.  33  y.  2  mo.  20  d. 

147.  86  y.  5  mo.  28  d. 
W.  60  days. 

150.  282d.  llh. 

152.  141wk.4d.  22I1. 

16  min. 

153.  37mi.  170rd.  1yd. 
154    lOT.  8cwt.  531b. 

155.  355  A.  49  sq.  rd. 
21f  sq.  yd. 

156.  203°  51'  40". 

158.  17 wk.  3d.   10 h. 

17  min. 

159.  3  mi.  124  rd.  1yd. 
11  ft. 

160.  3oz.  17pwt.  14gr. 

161.  13°  10'  35". 

162.  61  gal.  1  qt.  1  pt. 


163.  3779  in. 

164.  S  351.75. 

165.  6  T.  12  cwt.  6  lb. 

167.  98  T.  3  ctl.  10  lb. 

168.  19  rd.  6  ft.  6  in. 

169.  19  A.  136sq.rd. 
68  sq.  ft.  9  sq.  in. 

170.  786  days. 


171  2T.  7%  J  lb. 

172.  196 d.  Oh.  49m. 

173.  0.39625. 

174.  ]  GU  cd.  ft. 

175.  55. 

176.  $15.25. 

177.  78  y.  2  mo.  24  d. 

178.  Lose$10.78f 

180.  None. 

181.  547d.  20h. 

Art.  213. 

25.  151.845"^. 

26.  1663.70  ^ 

27.  17.4  ^^ 

28.  2040  \ 

29.  $527. 

30.  356.2075  "^ 

32.  121.2531b. 

33.  365.976  sq.  yd. 

34.  1111.95  acres. 

35.  88.9+^. 

36.  74+ ^^ 

38.  $1.84+. 

39.  $4637.60+. 

40.  2586.145+. 

41.  93.57"^. 

42.  $6.68+. 

43.  $922.36. 

Art.  222. 

6.  820  9375  sq.  ft. 

7.  7854  sq.  ft. 

8.  1256.64  ft. 

9.  400  ft. 

10.  31yd. 

11.  0.78125  yd. 

12.  13i  A. 

13.  154.56+ ft. 

14.  232  sq.  ft. 

15.  14.6770  «\ 

16.  $240. 

17.  41  sq.  rd.    147+ 
sq.  ft. 

18.  66.25  sq.  ft. 

19.  5  A.    159  sq.   rd. 
260i  sq.  ft. 


Area,128Msq. 
^0.  {      yd. 

Cost  $232.03+ 

Art.  226. 

25.  15  cu.  ft. 

26.  11571  cu.  ft. 

27.  3f  ft. 

28.  3  cu.  m, 

29.  793  J  cu.  yd. 

30.  8.700  cu.  m. 

31.  8.3776  cu.  yd. 

32.  1026  sq.ft.' 


Art. 

8c(i. 
8cd, 

^  ft. 

^4rd 


227. 


34. 
35. 
36. 
37. 
38.  $236.25. 

Art.  230. 

40.  U  bd.  ft. 

41.  66§  bd.  ft. 
4^.  283ibd.  ft. 
43.  $40,986. 
44'  $52.92. 


45.  72  ft. 

46.  $5.67+. 

47.  1220  bd.  ft. 
$7.68. 
24.75  "1. 
9  rolls. 
38.4  bu. 
65f  gal. 
8500  bricks. 
4  T.  and  31 T, 
7002  \\  lb. 
1782  cu.  ft. 


4S. 
49. 
50. 
51. 
52. 
53. 
54. 
55. 
56. 


Art.  231. 


50.  99. 

51.  0.003. 

52.  0.01875. 


N 


56 


356 

53.  0.42f 

54.  6400. 

55.  $132.37i 
(  $  10000  land. 
\  $  4000  house. 

57.  124  acres. 

58.  $2195. 

59.  42i|. 

60.  15  sq.  yd.  3  sq. 
ft.  128  sq.  in. 

61.  62f  yd. 

e^.  1613^  cu.  yd. 

63.  47520  bricks. 

64.  95040  bricks. 

65.  85.84  sq.  rd. 

66.  316i|^tons. 

67.  37i  sq.  yd. 

68.  $32.08^. 

69.  $11.50. 

70.  8739yV7. 

71.  32^1  cd. 
7^.  303  ^3_  sq.  ft. 

73.  74|i  yd. 

74.  5/y  acres. 

75.  $20.14ff. 

76.  $15.40. 

77.  0.5184. 

78.  0.625. 
7a  80  yards. 

5(9.  July  26,  11  h.  45 
min.  P.  M. 

81.  ^  sq.  yd. 

82.  $8.49f. 
5.5.  Ans.  If 

54.  88  sq.  rd. 
85.  23y\ft. 
5^.  18  ft. 

87.  1926  ft. 

55.  16|yd. 
5a  5832. 
90.  475  ft. 
Pi.  $12015 
92.  .Vir- 
95.  $351. 
P^.  $53.33f 

95.  95  d.  5  h.  10  min. 

96.  13500  yd. 

97.  yd.  wide;  $7.50. 
9d.  il  T. 


ANSWERS. 

99.  $364. 

57. 

1040  yd. 

100.  $2.25. 

55. 

3240  rd. 

101.  $r73.45yV 

89. 

$  3500. 

102.  im. 

90. 

7000  lb. 

103.  $0.88f 

91. 

$  166f. 

104.  $  0.52^. 

92. 

$  2400. 

93. 

$  9000. 

Art.  241. 

94. 

13655. 

35.  $3605. 
5^.  $315. 

Art.  246 

102. 

$1.55. 

Art.  242. 

103. 

$12160. 

37.  2380  tons. 

38.  205.20. 

39.  22.61  mi. 

104. 
105. 
106. 
107. 

$  50  loss. 
$10.44. 

$22.78^. 
$3780. 

40.  1211  men. 

108. 

$250. 

41.  $487.50. 
4^.  $51.64. 

109. 
110. 

6i%. 
41 A  %. 

4S.  $0.8136. 

111. 

17  ''' 

$3140. 

44.  $511.05. 

112. 

S490. 

45.  $4122.50 

113. 

$  8.34f . 
$1460. 
4  %  loss. 
$0.57^. 
$19  lOf. 

46.  $11223.87. 

47.  $9.34. 

114. 
115. 

48.  14748. 

49.  $  12825. 

116. 
117. 

50.  $5700. 

118. 

$66.66|. 

51.  $337.97. 

119. 

Neither. 

^i  91  %. 

Art.  248 

Art.  243. 

123. 

$141.95. 

^5.  %\%. 

124. 

$8.75. 

^^-  Mil 

125. 

$157.93. 

67.  25f  %. 

126. 

$59.80. 

68.  9|%. 

127. 

$5012.50. 

ea  83j%. 

128. 

$12876. 

7a  14%. 

130. 

$18.94. 

71.  5%. 

131. 

214Abbl. 

7^.  7i%. 

132. 

$1426.80. 

73.  15%. 

133. 

3.83+%. 

7^.  m%. 

134. 

$38.59. 

5^.  $324.40. 

135. 

$4170. 

83.  295  sheep. 

136. 

$200. 

Art.  244. 

Art.  252 

84.  $1562.50. 

139. 

$73. 

85.  14.4  tons. 

140. 

$124.76 

86.  66f  bu. 

141. 

$1531. 

ANSWERS. 


357 


IJ^.  $53625. 
1J,S.  $2400. 
lU'  2%. 
145.  $3075. 


14^-  m\i' 

147.  921%;  7^%. 
1J^8    36^V%- 
i^.  Lose  $  0.05. 
150.  110%. 
iJi.  $1097.25. 
152.  115  bbl. 

$115. 

$  26163.26. 

$5.31. 

$5f. 

$  361.80. 

$12375. 

$195. 
$  3033.75. 


153. 
154 
155. 
156. 
157. 
158. 
159. 
160. 
161. 
162. 

163.  $4600. 

164.  $  10872.75. 

165.  20%. 

1008.77H- 
42|  yd. 

$2.80ff  perbbl. 

5%. 

$  22790. 

195. 

33i%. 

$2.50. 

99J%. 

$4. 

298000. 


166. 
167. 
168. 
169. 
170. 
171. 
172. 
173. 
174. 
175. 
176. 
177. 


Art.  259. 

$2116.80. 
$12. 

$1728.75. 
$  343.931 
$584.6256. 

19.  $158.0298. 

20.  $8.2586. 
n.  $19.3909. 
22.  $125.71. 


12, 
13 
16 
17 
18 


34. 
35. 
36. 
37. 


23.  $299.33. 

24,  $6380.55f 

Art.  263. 

28.  $5.8409J. 

29.  $1500. 

30.  $  1.9958f . 

31.  $90.686f. 

32.  $35.7857i. 
""  $18.4255| 

$  30.87987^ 
$4339.40. 
$1610. 
$9205. 

38.  $15.986f. 

39.  $  1.3410^^. 

40.  $10.9450|. 

41.  $63.54. 

42.  $2358.90iV 
4^.  $140.5872. 
44'  $175. 

45.  $110.1012. 

46.  $818,545. 

47.  $  112.425. 

48.  $565,935. 

49.  $508.0765. 

Art.  264. 

52.  $1.2177. 

53.  $144.66|. 

54.  $1.30^. 

55.  $  101.25. 

56.  $  17.10. 

57.  1 167.96. 

58.  $962.6446. 

59.  $201. 

60.  $779.3264. 

61.  $4535  7355. 

62.  $4,774. 

63.  $446.25. 

64.  $26.9458. 

65.  $173.20. 

$  426.449. 

$637.19. 

$22,575. 

$  279.77505. 

$320.28. 


66, 
67 
68, 
69 
70 


71.  $178.87i. 


74 

75. 
76. 
77. 
78 
79, 


82. 


85. 


72.  $  131.15f . 

73.  $125,306. 
$  7.06446. 
$10.54928. 
$0.4216. 
$17.7818. 
$  569.201. 
$  900.2532. 

80.  $  1261.936. 

81.  $1025.814. 
$  860.57i 
$656.19776. 
$498.1983. 
$952,899. 

86.  $1968. 

87.  $968.66304. 

88.  $204.83^ 

89.  $901.6677. 

90.  $1237.88254. 

91.  $633.63554. 

Art.  265. 

93.  $45.62. 

94.  $16.52. 

95.  $236.73. 

97.  ^%. 

Art.  266. 

98.  6%. 

99.  6%. 
100.   61^^ 


%. 


101.  15f  i 

102.  7%. 

103.  ^\%. 

104.  9  %. 

105.  10%. 

106.  h\%. 

107.  4%. 

109.  6  years. 

Art.  267. 

110.  2  y.  11  mo.  28:^  d. 

111.  ly.  4  mo.  20(1. 

112.  2  mo.  6  d. 

113.  1  mo.  18  d.  or 
48  d. 

114.  2  y.  6  mo. 


358 


ANSWERS. 


115.  16  y.  8  mo. 

116.  7  y.  4  mo.  26|  d. 
m.  3y.  lmo.4^fd. 

118.  4y.  7  mo.  16fd. 

119.  9  y.  6  mo.  84  d. 
122.  $  75. 


Art.  268. 


123. 


\  3600. 
124.  1300. 
i^5.  |718.23fi-. 
i-^6.  ^14000. 

i^r.  $24000. 

128.  $675.60. 

129.  $14215.38yV 

130.  $14000. 

131.  $14400. 
182.  $36956250. 

Art.  282. 

134.  $394.57. 

135.  $448.35. 

136.  $492.06. 

137.  $342.40. 

138.  $1447.08. 

139.  $722.17. 
UO.  $5947.63. 
Ul-  $413.43. 
IJi^.  $1828.69. 
lJi3.  $1280.20. 

Art.  283. 

145.  $349.21. 

Art.  285. 

Ul.  $161.63. 

Art.  286. 

152,  $111.94. 

153.  $65.60. 
WJi,.  $93.70. 

155.  $1125.51. 

156.  $1273.8?. 

157.  %n%M. 

t.5$.  Sil.8$  gajij. 


Art.  287. 

161.  $1934.84. 

162.  $232.46. 

Art.  291. 

7.  $42. 

Art.  292. 

8.  $3122.17. 

9.  $28.81. 

10.  $356.44. 

11.  $130.32. 

12.  $300. 

Art.  294. 

13.  $6.25. 
U   $329. 

15.  $26.89. 

16.  $393.73. 

17.  $460.64. 

18.  $  20. 

19.  $213.82. 

20.  $734.25. 

Art.  300. 

( Bank    discount 
\      $17.94. 
I  Proceeds 
(.     $857.06. 
i  Bank    discount 
1      $3.05. 
)  Proceeds 
i      $82.55. 

Bank    discount 
$6.30. 

Proceeds 
$593.70. 

4880. 
$10635.40. 
$479.56. 
$953.23. 

30.  $1013.33. 

31.  $284.39. 

32.  $640.12. 

33.  $957.93. 
3J^.  $323,77. 
35.  $693.74. 


23^ 


25. 


27. 
28, 
29. 


37. 
38 
39. 
40. 


$  839.88. 
$494.92. 

$8567.37. 

$235.50. 

$795.07. 

41.  $555.27. 

42.  $568.96. 

43.  $888.85. 

45.  $1518.60. 

46.  $300. 

Art.  301. 

47.  $450. 
43.  $1600. 
4^.  $842.00+. 
50.  $509.34. 


51.  $24  discount. 

52.  $40. 

53.  April  15,  1878. 

54.  $6000. 

55.  Dec.  24. 

56.  $385.25. 

57.  $12.43. 

58.  $3.41. 

59.  $362.30. 

60.  April  4,  1882. 

61.  $175  loss. 

62.  $502.77. 

63.  $773.63. 

64.  $1722.64. 

•  Art.  314. 

8.  $2050. 

9.  $34087.50. 
10.  $1268.75. 

12.  $128.12f 

13.  $124,571^7. 

14.  $130. 

16.  $125. 

17.  $360. 

18.  Neither. 

20.  $21100. 

21.  $  16350. 

22.  $54500. 
^4-  4A%. 

25.  6  s,  .24%  greater. 

26.  6|  %. 


ANSWERS. 


359 


les,  $  85f 
^9.  At  133^. 

Art.  323. 

5   $1175.64. 

6.  $3900. 

7.  $2520.84|J. 
9.  $2517.69+. 

10.  $4000. 

11.  $  1164. 

12.  $450. 

U-  $  1474.12i. 
i5.  $2998.50. 
17.  $4000. 
i<?.  $500. 
iP.  $2985.067+. 

Art.  330. 

21.  £  160  8  s. 

22.  $  6437.43f . 

23.  $462.13tV3- 

24.  $289.0Ufi 

25.  16614. 

26.  $309.70. 

27.  5948i9^3-  marks. 

28.  $6117.03. 

29.  5434.03^  francs. 


7. 

8. 

9. 
10. 
11. 


Art.  334. 

March  18. 
Sept.  13. 
3  mo.  3  d. 
July  17,  1881. 
May  4. 


12.  46  days. 

Art.  335. 

IS.  Sept.  18,  1881. 
U-  Feb.  12,  1881. 

15.  July  1,  1881 

16.  Feb.  23,  1882. 

Art.  336. 

i7.  66f%. 

18.  $6740. 

19.  %  1053. 


20.  $585. 

21.  85%. 

22.  45Hf  %. 

23.  $12.25. 
^4.  $120.25ff. 

25.  $35.21+. 

26.  1  y.  2  mo. 
^7.  $600. 
28.  6%. 

^P.  8  V.  4  mo. 
30.  $69.79. 
5i.  $167,214-. 
cf^.  $40.04. 

33.  $1700.40. 

34.  $41745. 

35.  $1050. 
.^^.  $80. 
^r.  92  davs. 

38.  2^3_%  discount. 
59    $265.95. 
40.  24da^s. 


;^7. 


Art.  348. 

9. 


28.  15. 

^P.  40. 

30.  9. 

5i.  i. 

32.  8. 

55.  30. 

54.  75. 

Art.  349, 

36.  $127.50. 

37.  $91. 

38.  90  men 

39.  18  days. 


40. 


45 
^7. 


Art.  350. 

$632. 
$22.68. 
1363  miles. 
9  days. 
$3li. 
11:^  days. 
7  mo. 
$lli. 
41^  days. 


ji9.  77^2. 

50.  $33.91|. 

51.  96  men. 

52.  $39.40|. 

Art.  352. 

54.  216. 

55.  7\  years. 

56.  24  men. 

57.  16  acres. 

58.  ^. 

59.  $10i 

60.  208  miles. 
1800  tons. 
17^V  days. 
50ff  weeks. 
72  acres. 
3600. 
169X  cd. 
240. 


61. 

62. 


64. 

65. 
66. 
67. 

68.  $2.88. 

69.  8l|  ft. 

70.  $189^. 

Art.  356. 

A's  gain  $  504. 
B's  "  $  756. 
C's     "    $420. 

Art.  357. 

(  A's  gain  $  720. 
8.  {  B's     ''    $  990. 
(  C's     "  $  1080. 
f  A's  share 
I      $1137.50. 
g    J  B's  share 
1      $1365. 
I  C's  share 
L     $1722.50. 
.^    (Hall's  $312. 
^''-  (Bishop's  $416. 
r  B's  loss  1024bbl 

11.  ]  C's  "  154f  bbl. 
D's  ''  1024 bbl 
A's  share  $100. 

12.  <!  B's     "       $  75. 
C's     "    $126. 


360 


ANSWERS. 


fA's   gain  $84. 
^^    \  B's      "     $36. 
)  Flour  per  bbl. 
(.     $8. 

Art.  358. 

A's  loss 


15. 


16. 


17. 


18. 


19. 


20 


21. 


B's  loss 
$  224.32|f . 

C's  loss 
$316,701^. 

A  pays   $400. 

B      "      $480. 

C      "      $450. 

A  pays  $24.75. 

B    "'     $22.05. 

A's  loss  $71f 

B's    "    $1334. 

C's    "    $95^\. 
CA's  gain  $960. 
(B's     "  $1200. 
(  A's  gain 
)      $1244f 
I  B's  gain 
l     $1155f. 

Davis's    profits 
$720. 

Wood's   profits 
$  1440. 

Furbusli's  pro- 
fits $  1000. 

A's  gain  $2200. 

B's    "    $1650. 

C's    "    $2000. 

Art.  363. 


8. 

9. 
10. 
11. 
12. 
.13. 
U. 
15. 


529. 

4096. 

28561. 

A- 

]5|. 

12.96. 

15625. 

161051. 

16.  0.003375. 

17.  12.25. 

18.  203^V- 

19.  0.000343. 


Art.  375. 

26.  96. 

27.  165. 

28.  427. 

29.  847. 

30.  974. 

31.  2.59. 

32.  2.05. 

33.  43.2. 

34.  0.097. 

35.  0.237+. 

36.  0.97. 

37.  279. 

38.  4.03. 
89.  3.86. 
40.  6.25. 
4i.  23.19. 
Jf2.  8.426. 
.^5.  0.0447. 

Art.  376. 

47.  7f. 

4^.  6f 

.^P.  7.031+. 

50.  M 

5i.  8^. 

52.  0.96+. 

5^.  1.5. 

5.4.  31J. 

55.  12.108+. 

56.  9.019+. 

57.  0.935+. 
5^.  0.829. 

59.  476  men. 

60.  49  rd. 

Art.  384. 

64.  45. 

65.  75. 

66.  83. 
(57.  9.7. 
68.  156. 
6.9.  235. 
7^.  39  2. 
71.  467. 


7^.  637. 

73.  18.54. 

75.  40.1. 

76.  334. 

77.  0.012. 
7^.  123. 
79.  2.012. 
^^.  0.423. 
^i.  0.205, 
82.  0.019. 
^<5.  2.92+. 

Art.  385. 

85.  lA- 
^6.  3|. 
^7.  0.84+. 
.5<5.  0.57. 
89.  4.33. 
m  5.503+. 
Pi.  25.81+  inches- 
Art.  390. 

3.  25  ft. 

4.  85  ft.^ 

5.  60  miles. 

6.  387.30  ft. 

7.  53.81+  id. 
^.  44.94+  ft. 

Art.  394. 

9.  540  sq.  ft. 

10.  192  sq.  ft. 

11.  199  sq.  ft. 

Art.  396. 

12.  9350  sq.  ft. 

13.  3  sq.  m. 
H.  Si  a. 

Art.  398. 

16.  1040  sq.  ft. 

17.  87iMi  -q.  yd 

Art.  400. 

18.  90  cu.  ft. 

19.  216000^"^"^. 

20.  153^V?cu.  ft. 


ANSWERS. 

361 

Art.  403. 

65.  7|  years. 

($375  A's. 

21,  69.825  cu.  ft. 

66.  $1251.99. 

105.  ^$625  B's. 

22.  62.42  cu.  m. 

67.  $  100.00+. 

(  $  1000  C's. 

23.  80041500  cu.  ft. 

68.  $  1539.75. 

106.  407.29+ in. 

69.  $  811.08+. 

107.  1.04. 

Art.  405. 

70.  $17968.75. 

108.  42.5  ft. 

25.  169.74+  cu.  ft. 

26.  61iV^  cu.  ft. 

71.  $1279.30. 

72.  $  1653.69+. 

73.  2.5. 

109.  22j\  gal. 

110.  5  mo.  21  d. 
(  First  123. 

Art.  407. 

74.  282.0638. 

75.  41  cents. 

111.  <  Second  37 
(  Third  22. 

27.  1963.5  sq.  in. 

76.  12|oz. 

112.  300  sq.  yd. 

113.  27. 

28.  0.101787+sq.m. 

77.  $3.35^. 

29.  65450  cu.  ft. 

78.  1484.84M. 

30.  4849059600 cu.mi. 

'  Ames 

Art.  414. 

$5165.91+ 

2.  1020. 

Art.  409. 

Stevens 

3.  8. 

4.  1125. 

5.  4. 

32.  32  sq.  ft. 

33.  100  rd. 

79.  < 

$3874.43+. 
Conant 

34    5890.5  sq.  ft. 
35.  -^1^  hours. 

$2152.46+. 
Hubbell 

e.  $41G6f. 
7.  365. 

36.  $67.50. 

.     $807.17+. 

8.  4. 

37.  949.95  sq.  m. 

80.  i 

S  313.532. 

9.  24. 

81.  127.279+  ft. 

i6).  744  hours. 

Art.  412. 

82.  452.54+ rd. 

39.  4.57+  ft. 

83.  29.6. 

84.  2  m.  66.274  rd. 

Art.  415. 

1.  18.    - 

40.  2.5  in. 

85.  16  ft. 

41.  132  K. 

42.  64  times. 
4S.  472i  lb. 

86.  74.83+ in. 

87.  208.71+  ft. 

88.  $  240  or  $  330. 

2.  365. 
.5.  1776. 

4.  18  days. 

5.  7677. 

7.  1248480000. 
5.  $330. 
9.  3102AV 

io.  19. 

Art.  413. 

49.  Neither. 

89.  38.1575  ft. 

90.  79.1959. 

91.  373218. 

50.  \2\%. 

92.  23.32+ ft. 

51.  $28.80. 

93.  1. 

52.  f,. 

53.  1176  sq.ft. 

94.  42336  sq.  in. 

95.  24. 

Art.  416. 

54    192^%%. 

9^.  91f 

4-  99084.19. 

55.  $9000. 

97.  2  :  5. 

5.  2708.71. 

56.  16f  cents. 

95.  $3.64. 

6.  8511. 

57.  $16644. 

99.  72. 

7.  2325. 

58.  $8382.40. 

iOO.  $2666.6Gf. 

f- 1^3^' 

59.  2000001b. 

(  20  days  at  the 

9.  $26. 

60.  5^\%. 

101.  <      distance      of 

ia  295  and  $15  over. 

61.  $1035.1657+. 

(      500  miles. 

62.  Lose  $  50. 

102.  55  min.  2^^  sec. 

Art.  417. 

63.  $88.73+. 

i(>5.  36  men. 

1.  5^9203200. 

64.  $84,906. 

104. 

67^  days. 

2.  9600  cu.  ft. 

362 


ANSWERS. 


3.  825. 

4.  $2.25. 

5.  2284.47. 

6.  24  lb. 

7.  717H. 
9.  $361. 30. 

10.  $12.24. 

Art.  418. 

i.  29S||. 
;^.  $113f 
.5.  8656742. 

4.  $885.50. 

5.  $  7.23. 

6.  $135. 

7.  $229.50. 
^.  14400. 

9,  Lose  $  703. 
10,  18090100. 

Art.  419. 
1.  424. 

6.  23,; 

7.  79ii;  19||. 

P.  210«f;  857;  85. 
10.  j\;  $100000. 

Art.  420. 

^.  14. 
3.  280. 

-a 


9 

14'    fV 


6. 


(37. 

ha*- 


9.   i\)\}^  miles. 
-/(?.  13J  bushels. 

Art.  421. 

1.  12. 
^.  360. 


5.  95J^V 
^-  41^. 
5.  306H. 

^.  H;  iVe- 

^.  $65621 
9.  146-^^  miles. 
10.  46f  tons. 

Art.  422. 

^.  450. 

^-  m- 
5.  20if. 

^-  m- 

7.  $  94500. 
5.  $4800. 

Art.  423. 

■'■•     5  J    5  >    T' 

2.  16ij%. 

5.  isH- 

.4.  2685f|. 
5.  3445f. 

^.  Hf^* 

7.  8^Vo- 
^.  $ff 

^-  II*- 
10.  80|  miles. 

Art.  424. 

^-  32H!;  8AV 
.^.  187|  miles. 
^    4fff. 
5.  $14000. 
6'.  $57600. 
7.  $12000. 

^-  iff- 
9.  50  acres. 
!(?.  $  109714f . 

Art.  425. 

1.  153^V 

2.  94jf. 

5     31-i+. 

.4.  $5640. 
5.  8ff . 


^.  6  lots. 

^.  t\. 
5.  f. 

9.  15)5625. 

i6>.  Diminished  ^. 

Art.  426. 

if- 

«^-  y58M. 

5.  |. 
^.  35. 

7.  $lf. 

^-   21fi;28f;6i|J. 
i^.  $  I841f .         ^^ 

Art.  427. 

i-  0.1875. 
^.  0.79992, 

(5.  24.5  yd. 

7.  1.6075. 

8.  $330. 
P.  S  28.26. 

10.  0.5. 

Art.  428. 

i.  0.009125. 

.^.  1.44. 

.4  0.0001177. 

5.  207.36. 

e.  0.01010625. 

7.  0.00125. 

P.  15. 

10.  479.9975, 

Art.  429. 

1.  0.003. 

2.  998998.999. 
S.  0.00144. 

Jf,.  166. 

5.  $40,937. 

6.  7  years. 

7.  $300. 

^.  $334.21875. 


ANSWERS. 


363 


9.  $49.81f 
10.  $9.06^. 

Art.  430. 

1.  10.2485. 

3.  0.72727 j\;  ^V 

4.  0.20445+. 

5.  6    oz.    11     pwt. 
8.448  gr. 

6.  $165. 

7.  $6702.24^. 

8.  16455  ^ 

P.  13  lb.  lA  oz. 
ia  39.37+  m. 

Art.  431. 

1.  283  y.  8  mo.  23  d. 

2.  50  A.  105  sq.  rd. 
S.  578   mi.  286  rd. 

3  ft. 

4.  88°  45'. 

5.  936if  lb. 

6.  58°  28'20''W.. 
^.  2  d.  0  h.  44  min. 

9.  2.32+sq.  rd. 
10,  338   cu.  ft.  1584 


Art.  432. 

1.  llf  rolls. 

2.  BiH-         , 

5.  7  gro.  2f  doz. 
4.  1680  sq.  in. 

Feb.  6. 

$  8.04f . 

353.7if. 

3AA. 
25 


$250. 


10.  25  sq.  yd. 

Art.  433, 

1.  $132. 

2.  $37.80. 

3.  13ift. 
^.  $6384. 

5.  10.84  oz. 

6.  $350.62i. 


<?.  53  bags. 

9.  1880  d.  6  h. 

10.  6  lots. 

Art.  434. 

1.  $5940. 

^.  37i  cents. 

3.  21780  cu.  ft. 

4.  3||. 


P. 


87H  ft. 
59319  bricks. 

-'■-'^T9200-       „ 

C  5600  bd.  ft. 

($89.60. 

$15. 


10.  $75.79. 


8. 


ft. 


Art.  435 
1.  1  A. 

^.  1609.3+ . 

3.  19350^^. 

4.  $6.43f. 

5.  $560. 
7.  160  ft. 

(  4860  bd. 
($48.60. 
9.  $60. 
m  $17817.18f. 

Art.  436. 

1.  34f4  yd. 

2.  154  cu.  yd. 

3.  29yV  mi. 

4  0.1090+  mi. 

5.  4873if|4. 

6.  $488,821 

7.  96    mi.    154   rd. 

8.  lisSU  lb. 

9.  71°0'W. 
'$58.07ijcostof 

the  $2  kind. 

$38.11^  cost 
of  tlie  $1.75 
kind. 

The  $2.00  kind 
the  more  ex- 
pensive. 


10.  < 


Art.  437. 

1.  48  rd. 

2.  600  sq.  rd. 

3.  351  T. 
.4.  18  min.  40  sec. 

5.  $1756. 

6.  $313.50. 

7.  $20.25. 

8.  0.6235+. 

9.  $25.68+. 
10.  1830  cu.  ft. 

Art.  438. 

14%;  44%;  40%; 
87i%;  60%; 
71f  %;  62i%. 
4-  45%. 

5.  $4331.25. 

6.  135°. 

7.  $14400. 

8.  50%. 

a    C$11955.17. 
^'  ($44.83. 
10.  $9000. 

Art.  439. 

1.  $2879.374- 

2.  $31. 

^    C$44231.71. 
^'  ($1105.79. 
.^.  $26.25. 
5.  75%. 

^     3  FIT- 

^-   ($44000. 
7.  38||i%. 
^.  $10666.66f 

9.  $89.63+. 
.0    5  $2784. 
^^-  J  $2876.80. 

Art.  440. 

i.  67i%. 
•^.  66f%. 
^.  $244.44+. 

4.  $40000. 

5.  20%. 


364 


ANSWERS. 


2. 
3. 

5. 


10. 


6.  Loss  $25. 

7.  6i%loss. 

5.  90%. 
9.  $4. 

10.  $2.80. 

Art.  441. 

A's$  43750. 

B's$  56250. 

C's  $  25000. 
20%. 
$50800. 

$4390.24. 

$  109.76. 

3600. 

6.  63^  cents. 

7.  $149,334. 

8.  $33.75. 

9.  $102.57. 
On  wheat. 
5  %  greater. 


Art.  442. 

1.  2376  ft. 

^-  42f%. 
S.  $20000. 
^  §3520. 
5.  $13.       ' 

^.  8AV%. 

7.  $32000. 
^.  11200. 
Q    U1740. 
$1832.80. 


If.   5  $56250. 
($2531.25. 

Art.  443. 

1.  60  cents. 

2.  $20000. 

.5.  Renting,  by  $  50. 

4.  $760. 

5.  12i%. 

6.  $24375. 

7.  $124. 
^.  $7.35. 

p   j  Paid  $300. 
"J  Received  $  336. 
10.  Gained  21%. 


Art.  444. 

1.  $153.51+. 

2.  12  y.  6  mo. 

3.  3520. 

4-  $89,359+. 

5.  $2.66f. 

6-  1  J.  8  mo.  12  d. 

7,  $640.51. 

^  $432.84. 

9.  $825.17. 

i^.  $0.04. 


i. 

2. 
3. 

4. 
5. 

9. 


Art.  445. 

3j.  2mo.  27+d. 

$6936.09. 

7%. 

$40.23. 

$169.82. 

($1171.87ipres- 

)      ent  worth. 

)$103.12i    dis- 

(      count. 

$508.35. 


10.  $1313.56. 

Art.  446. 

1.  $6.98. 

2.  $13.56. 

3.  $366.26f. 

4.  5|%. 

5.  $904.97. 

6.  $190.40. 

7.  $28.57. 

8.  $225. 

r  $9.30  bank  dis- 
Q   )      count. 
"^^  1$  440.70      pro- 

V     ceeds. 
10.  $251.36. 

Art.  447. 

1.  $13,697+. 

2.  2  y.  7  mo.  15  d. 

3.  8%. 
4-  $300. 
5.  $5.55. 


^.  $2440. 

7.  $550.87^ 

8.  $154.61. 

9.  $1688.26. 
10.  $953.49. 

Art.  448. 

1.  9  V.  6mo.  9d* 

2.  llf  %. 

3.  $1440.831 
4-  $0,269. 

5.  $303.97. 

6.  $  210. 

7.  $400.07. 

8.  $742. 

9.  $4375. 

i6>.  Lessen  If  %. 

Art.  449. 

1.  5  7'. 

2.  $40000. 
^.  24f  %. 

■4.  19ff  %  gain. 

/-    (151  shares. 

■   ($122f  left. 

6.  1\  mo. 

7.  19  mo. 

^.  $2432.62+. 
9.  Sept.  22,  1880. 
ia  Feb.  6,  1882. 

Art.  460. 

2.  9  yds. 

^.  m  ft. 

^.  9X  days. 

5.  T\^  oz. 

6.  22^^  acres. 

7.  h\  mo. 
<?.  96  men. 
9.  102  men. 

10.  6  men. 

Art.  451. 

1.  $2000. 

1st  man  $15. 

2d  man  $20. 

.3d  man  $20. 


ANSWERS. 


365 


5. 


a.   5  Ames  $1600. 
•^^  J  Howe  $1200. 

!  A  $62.50. 
B$75. 
C$100. 
D  $112.50. 
A's  $  5333J-. 
B's$66G6f. 
(A's$388fif. 
6.  ^B's$2493Vt. 
(C's$  112^3^,-. 
;y    j  1st  mau  $  900. 
^'  \  2d  man  %  600. 
(  A's  %  3516.80. 

8.  I  B's$5861.33J. 
(C's$8205.86f. 

9.  2||  days. 

r  37  %. 

A  $666. 
B$  277.50. 
C$721.50. 


Art.  452. 

962. 

36.37. 

93.27  ft. 

$ 187i. 

0.199999488. 

21^9^  rd. 

324. 

12  minutes. 

8.55—. 

r  Slant  li't  100  ft. 

Surface    24000 
sq.  ft. 

Contents 

384000  cu.  ft. 


to. 


1. 


to.^ 


Art.  453. 

1.  $133J. 

2.  72  in. 

3.  1G4  ft. 

4.  46.596+  rd. 

5.  307.3-1-  miles. 
'24576    sq.    in. 

entire  surface. 
'  110.8  in.  diago- 
7.  27.  [nal. 

S.  8  sq.  ft. 


Ml 


9.  63.24  ft. 
10.  6^1^  ft. 

Art.  454. 

2.  0.0214f. 
^.  $28571. 
.4.  $172.46. 

r$  22.60       bank 
;-   3      discount. 
^-  1  $  1227.40    pro- 

V.     ceeds. 
6    4  li.  48  min.  p.m. 

7.  $151.25. 

8.  $187.20. 

9.  42. 
10,  9Jf  d. 


Art.  455. 

76  yd. 
$  242.92. 
128  rd. 
3|ft. 
0.000002. 
183^. 
$10. 
$91.80. 
64  rd. 


10.  14  days. 

Art.  456. 

1.  35000  times. 


3. 


16, 


75000. 
5%. 
$8.16. 
Due    May 

1881. 
Bank    discount 

$19.37i 
Proceeds 
$1230.62^ 
days. 
20 'rd. 
452.39  cu.  in. 


5A< 


Art.  457. 

1.  $0.91. 

2.  25y%  acres. 


3.  20%  gain. 

4.  4166|. 

5.  $202.0095. 

6.  Gain  $66.98. 

7.  274f  lb. 

8.  900  sq.  ft. 

9.  2y.  8mo.  28d. 
10.  152+ ft. 

Art.  458. 

1.  175  sheep. 

2.  $3.81f. 

3.  80  lb. 

i  $360    share   of 
,    )      1st. 
•^-  ^$640   share   of 

(      2d. 

5.  $8.46. 

6.  174.4+. 

7.  45  min. 

8.  $2000, 

9.  8%. 
10.  5%. 

Art.  459. 

1.  $476. 19^^. 

2.  $3885120. 

3.  1.15^V^  mi. 

5.  $100. 

(One  17-i°webi 

6.  I  The   other   30° 
(      east. 

7.  $  ]  9f  ^  gain. 

8.  $57600. 

5>.  460  sq.  rd. 
10.  364f|  %. 

Art.  460. 

1.  $2640. 
^.  15  ft. 
^-    It^A; 

4.  28f  ft. 

5.  $449fiHgain. 

6.  $4442.81  J. 

7.  $452.17^- 
5.  36  men. 

a  226.27  rd. 
10.  338ii  cu.  ft. 


366 


ANSWERS. 


Art.  461. 

1.  $867.94. 

2.  12|i 

S.  $i:3.516. 

5.  l^V 

6.  1898^ViVlb. 

7.  B. 

8.  1421.2296  sq.  ft. 

9.  B  ^%  greater. 
10.  0.105+. 

Art.  462. 

1.  $32.56J^J. 

2.  17280. 
S.  $5.04. 

4.  $6726.56i. 

5.  $17.22. 

6.  $112.50. 

7.  31f 

8.  $184.129|. 

9.  $1636. 
10.  $2969.30. 


Art.  463. 

1.  $5908. 

2.  $27.59. 

5.  At  18  months. 

4.  31050. 

5.  $312.50 

6.  40.84+  ft. 
r$  15.75     A's 

share. 

$14.62i   B's 

share. 

'  $18.37 J    C's 

share. 

$26.25     D's 

sliare. 
$816|C'ssliare. 
$  1066f     B's 

share. 
$1116§     A's 
share. 
24  ft. 
29f^  minutes. 


7. 


S.  < 


9. 
10. 


16. 
17. 
18. 
19. 

20. 


Art. 

Iff 
If- 


-W(7^-. 


474. 


:^^.  4.21295. 
;^.^.  0.476. 

24.  10.083. 

Art.  478. 

25.  1.509375  mile. 

26.  76.835  A. 

27.  17.545  A. 

28.  575  miles. 

29.  486. 
.^^.  5760. 
31.  $225. 
<^^.  $34. 

Art.  486. 

36.  5h.8min.llfsec. 

37.  10  h.  53  min.  37| 
sec. 

38.  1*2°  27'  14''. 
^P.  13°  22'  30". 
40.  93°  40'  W. 

Art.  493. 

44.  $768.12. 

45.  1435.41. 
4<5.  $707.54. 

Art.  494. 

48.  $1179.04. 

49.  $4584.33. 

Art.  497. 

51.  Aug.  7,  1881. 

52.  Nov.  12,  1881. 

53.  Feb.  6 

5^.  Dec.  16,  1881. 

Art.  505. 

56.  $32.29. 


57. 


58.  -{ 


C's$  116.28. 
D's  $22.69. 
E's$  69.06. 
fEachpoll$1.58. 
County        rate 

0.0009. 
Town  rate 

0.0121. 
A's  county  tax 

$5.50. 
A's    entire   tax 
L     $66.58. 

Art.  510. 

(  $  1400  duty. 

59.  I  $  4929.02+ 
(      cost. 

60.  $1851.29+. 

61.  $2532f 

62.  $5105.979+ 

63.  $182.68+. 

Art.  614. 

64.  22.96+ cu.  ft. 

65.  37i  cu.  ft. 

Art.  515. 

66.  14.847  in. 

67.  431.869+ bd.  ft 

68.  $13.12+ 

Art.  518. 

69.  49.364  gal. 

70.  116.28  gal. 

71.  369.309+ L 

72.  446.25  gal. 

Art.  520. 

73.  119Xtons. 

74.  672lf  tons. 

Art.  526. 

75.  764  bu. 

76.  22|  bu. 

77.  8640  lb. 

78.  5JJbu. 

79.  $  15. 


ANSWEKS. 


367 


80. 


81. 


'186  Ib.^  rump 
and  sirloin. 

'  186  lb.  round. 

'4501b.  hams  & 
shoulders. 

'  600  lb.  sides. 


Art.  532. 

82.  ^5.71f 

83.  $51.02f. 

84.  $35.0S^V 

85.  $179.64. 

86.  $362.20+. 

Art.  538. 


87. 


88. 


89. 


90. 


91. 


15.888  M.  shin- 
gles. 

95.3281b.  nails. 

$19.86  cost  of 
labor. 
14.53  hundred. 

!$2.47   cost    of 
paint. 
$  1.78  paid 

painter. 
(  20|-      hundred 
1      laths. 
^$0.64-|-      cost 
{      of  nails. 
(  $  76     cost     of 
1      boards. 
)$4.27ito  each 
(      man. 


Art.  543. 

Pf  302. 

95.  23. 

96.  9  miles. 

97.  65  cents. 

Art.  544. 

99.  20640. 
100.  $348. 

Art.  546. 

103.  1600000. 
10^.  $142.92+. 
105.  $63.12+ 


Art.  547. 

107.  1530. 

108.  30^^. 

109.  437.46+. 

110.  $3495.25. 

Art.  548. 

1.  Eight       ten-thou- 
sandths. 

2.  $29741.25. 

3.  39"'. 
4-  20%. 

Art.  649. 

2    jSA^^sum. 
( Ti  pi'oduct. 

3.  1566.7+. 

4.  1880.0001+. 

5.  488.25  +  K. 

e.  36if  %. 


Art.  550. 

Eighty-five  mil- 
lions eighteen 
thousands  nine 
hundred  eighty- 
two. 

9^. 

5  500.002. 

(499.998. 

5  Kg  217  s^. 

13.572-f-. 


1. 


Art.  551. 

63. 

2.  0.555f 

3.  2.17+ "^ 

4  59^Vyd. 

5.  20  %. 

Art.  552. 

1.  0.64+. 

2.  4.69+ 

3.  O.OSO^J  mile. 
4  36,5+^'". 


Art.  553. 

1.  2. 
L.  C.  M.  360. 
G.  C.  D.  2. 

3.  7.62 '». 
4    1879. 

5.  $0.88f 

6.  $1142f 

Art.  554. 

2.  $12520.24. 

3.  1.6093+ Km. 

4-  i,  or  .5. 

i  Spent  $360. 
5.  I  Saved  $240. 

(Tobusiness$180. 
^.  If  years. 

7.  8  %. 

8.  0.899. 

Art.  555. 

3.  0.577+. 

4    0.024||f. 

5.  0.8314^^5^  acre. 

neo^^dm. 

•   [300000. 

Art.  556. 

1.  7. 

2.  67.2^™. 

3.  5040. 

4.  9  %. 

5.  26f  days. 

Art.  557. 

1.  1234567.654321. 

2.  1595864. 

JLXXXIVDCG- 
XCVI. 

Twenty  millions  five 
hundred  sixty- 
seven  thousands 
one  hundred 

eighty-nine,  and 
four   million  three 

-  hundred  twenty- 
one  thousand  nine- 
ty-eight billionths. 


368 


ANSWERS. 


6.  330377111!  J. 

6.  129. 

7.  810. 

10.  87. 

il.  $654.50. 

12.  35°  30'  eastward. 


13.  iofl%. 
i^.  525. 
i5.  $413.43f. 
16.  4|f  hours. 
i7.  $5.87. 
18.  $289.76. 
iP.  $125.03. 
20.  6. 


;^^. 
24. 
25. 
26. 


A's$640. 

B's  $  840. 

C's  $  840. 
211  tons. 
0.062. 
140.5  miles. 
%  19.44. 
19.635  sq.  ft. 


27.  20  sq.ft. 


TTSE 


I 


^% 


^r-  ss 


/ 


?    2^ 


-^i ^   ■« 


918313 


THE  UNIVERSITY  OF  CALIFORNIA  UBRARY 


